Indeterminacy of Boolean Ring

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Ahmed" }, { "@type": "Person", "name": "Majid Mohammed Abed" } ], "publisher": { "@type": "Organization", "name": "F1000Research", "logo": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 480, "width": 60 } }, "image": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 1200, "width": 150 }, "description": " Background A neutrosophic ring represents an algebraic generalization of the classical ring structure by introducing an indeterminacy element I , enabling the modeling of truth, falsity, and indeterminacy simultaneously, as established within Smarandache’s neutrosophic framework. In contrast, a Boolean ring is a commutative algebraic structure in which every element is idempotent ( αI ) ² = αI reflecting the logical principles of Boolean algebras and possessing characteristic two Combining these concepts, the neutrosophic Boolean ring extends the Boolean ring by embedding neutrosophic logic parameters—truth (T), indeterminacy (I), and falsity (F)—into its elements and operations. This hybrid structure allows for the representation of algebraic uncertainty and incomplete information while preserving Boolean idempotent properties, thus providing a flexible framework for studying systems with uncertain or partially defined information in algebraic and logical contexts Methods The research defines the Indeterminacy ring R I = { α + βI : α , β ∈ R } and explores its algebraic properties through examples from integers, rationals, and reals. It then formulates the Indeterminacy Boolean Ring (B-Ring) characterized by idempotency ( αI ) ² = αI , and establishes several theorems proving its core algebraic features. Results Findings reveal that Indeterminacy B-Rings are commutative and have characteristic two, ensuring 2 αI = 0 . Each maximal Indeterminacy ideal is also prime, and these rings are semisimple and reduced, containing no nonzero nilpotent elements. Furthermore, any Indeterminacy B-Ring can be represented as a direct product of copies of Z ₂ I , known as the Indeterminacy Boolean field. The quotient rings preserve Boolean and Indeterminacy properties, confirming their structural consistency. Conclusions The study successfully extends Boolean ring theory to the Indeterminacy domain, establishing a strong algebraic foundation for modeling uncertainty. Indeterminacy B-Rings maintain the essential Boolean properties of idempotency and commutativity while incorporating indeterminate behavior through I ² = I . These results open new perspectives for future applications in neutrosophic logic, fuzzy systems, and abstract algebra dealing with indeterminate information. " } { "@context": "http://schema.org", "@type": "BreadcrumbList", "itemListElement": [ { "@type": "ListItem", "position": "1", "item": { "@id": "https://f1000research.com/", "name": "Home" } }, { "@type": "ListItem", "position": "2", "item": { "@id": "https://f1000research.com/browse/articles", "name": "Browse" } }, { "@type": "ListItem", "position": "3", "item": { "@id": "https://f1000research.com/articles/15-205", "name": "Indeterminacy of Boolean Ring" } } ] } Home Browse Indeterminacy of Boolean Ring ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article Ahmed YA and Mohammed Abed M. Indeterminacy of Boolean Ring [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :205 ( https://doi.org/10.12688/f1000research.172934.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article Indeterminacy of Boolean Ring [version 1; peer review: 2 approved with reservations] Yousif A. Ahmed https://orcid.org/0009-0000-5308-0356 1 , Majid Mohammed Abed 1 Yousif A. Ahmed https://orcid.org/0009-0000-5308-0356 1 , Majid Mohammed Abed 1 PUBLISHED 06 Feb 2026 Author details Author details 1 Mathematic, University of Anbar, Ramadi, Al Anbar Governorate, Iraq Yousif A. Ahmed Roles: Writing – Review & Editing Majid Mohammed Abed Roles: Writing – Review & Editing OPEN PEER REVIEW DETAILS REVIEWER STATUS This article is included in the Fallujah Multidisciplinary Science and Innovation gateway. Abstract Background A neutrosophic ring represents an algebraic generalization of the classical ring structure by introducing an indeterminacy element I , enabling the modeling of truth, falsity, and indeterminacy simultaneously, as established within Smarandache’s neutrosophic framework. In contrast, a Boolean ring is a commutative algebraic structure in which every element is idempotent ( αI ) ² = αI reflecting the logical principles of Boolean algebras and possessing characteristic two Combining these concepts, the neutrosophic Boolean ring extends the Boolean ring by embedding neutrosophic logic parameters—truth (T), indeterminacy (I), and falsity (F)—into its elements and operations. This hybrid structure allows for the representation of algebraic uncertainty and incomplete information while preserving Boolean idempotent properties, thus providing a flexible framework for studying systems with uncertain or partially defined information in algebraic and logical contexts Methods The research defines the Indeterminacy ring R I = { α + βI : α , β ∈ R } and explores its algebraic properties through examples from integers, rationals, and reals. It then formulates the Indeterminacy Boolean Ring (B-Ring) characterized by idempotency ( αI ) ² = αI , and establishes several theorems proving its core algebraic features. Results Findings reveal that Indeterminacy B-Rings are commutative and have characteristic two, ensuring 2 αI = 0 . Each maximal Indeterminacy ideal is also prime, and these rings are semisimple and reduced, containing no nonzero nilpotent elements. Furthermore, any Indeterminacy B-Ring can be represented as a direct product of copies of Z ₂ I , known as the Indeterminacy Boolean field. The quotient rings preserve Boolean and Indeterminacy properties, confirming their structural consistency. Conclusions The study successfully extends Boolean ring theory to the Indeterminacy domain, establishing a strong algebraic foundation for modeling uncertainty. Indeterminacy B-Rings maintain the essential Boolean properties of idempotency and commutativity while incorporating indeterminate behavior through I ² = I . These results open new perspectives for future applications in neutrosophic logic, fuzzy systems, and abstract algebra dealing with indeterminate information. READ ALL READ LESS Keywords Indeterminacy ring; Prime ideal; Maximal ideal; Idempotent; Indeterminacy Boolean ring; semi simple; homomorphism; Indeterminacy Boolean field Corresponding Author(s) Yousif A. Ahmed ( [email protected] ) Majid Mohammed Abed ( [email protected] ) Close Corresponding authors: Yousif A. Ahmed, Majid Mohammed Abed Competing interests: No competing interests were disclosed. Grant information: The author(s) declared that no grants were involved in supporting this work. Copyright: © 2026 Ahmed YA and Mohammed Abed M. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The author(s) is/are employees of the US Government and therefore domestic copyright protection in USA does not apply to this work. The work may be protected under the copyright laws of other jurisdictions when used in those jurisdictions. How to cite: Ahmed YA and Mohammed Abed M. Indeterminacy of Boolean Ring [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :205 ( https://doi.org/10.12688/f1000research.172934.1 ) First published: 06 Feb 2026, 15 :205 ( https://doi.org/10.12688/f1000research.172934.1 ) Latest published: 06 Feb 2026, 15 :205 ( https://doi.org/10.12688/f1000research.172934.1 ) 1. Introduction Fuzzy theory is one important of many branches in mathematic. Many authors have investigated indeterminacy-based algebraic structures. In particular, Smarandache introduced the general framework of (T, I, F)- Indeterminacy structures and explored their algebraic properties. 1 In Agboola, 2 several fundamental results in lattice theory were developed, forming a structural basis for algebraic systems involving order relations and ideal theory. These foundations can be extended to study the behavior of neutrosophic and indeterminacy-based lattices, where it is proved that in Indeterminacy B-Rings every maximal ideal is also a prime ideal (Chalapathi and Madhavi). 3 More results related to idempotent elements have been presented in Al-Hamido, A., 4 The property of idempotency plays a central role in Boolean and Indeterminacy B-Rings. Ali and Smarandache 5 presented a comprehensive survey of neutrosophic and indeterminacy-based algebraic systems, outlining the general framework of Indeterminacy algebra. Later, Chalapathi and Madhavi further developed the structural aspects of Indeterminacy B-Rings. Also, in, 15 – 17 some information about fuzzy ideal and some definitions in Indeterminacy theory. In, 6 , 7 the authors presented an integrated framework of Indeterminacy set. The existence of a multiplicative identity e=I in (R ∪ I) guarantees structural stability and allows generalization to broader algebraic contexts. 3 Additionally, illustrative examples were presented to highlight cases where idempotency holds or fails, alongside remarks connecting algebraic logic with the structural properties of Indeterminacy B-Rings. 8 Thus, this research focuses on Indeterminacy groups, maximal ideals, prime ideals, idempotent, and especially Indeterminacy B-Rings, presenting new results and extending the existing ones to enrich the field of Indeterminacy algebra These insights pave the way for further applications in fuzzy mathematics and the theory of ideals. Finally, in the context of algebraic logic, the connections between Indeterminacy rings, Indeterminacy groups have been investigated to highlight the structure of Indeterminacy evaluation rings, their commutativity, and their role in lattice theory. 2. Methods Note: Throughout this article, the notation “ R I ” will be used in place of R I Definition 2.1. 11 Consider S as a non-empty set. An Indeterminacy set A I on X is known as: A I = { ( α , T A ( α ) , I A ( α ) , F A ( α ) ) : α ∈ S } , were T A ( α ) ⊆ [ 0 , 1 ] , (Truth-membership), I A ( α ) ⊆ [ 0 , 1 ] , (Indeterminacy membership), F A ( α ) ⊆ [ 0 , 1 ] (Falsity membership). of an element x in the set A I . Remark 2.2. The fundamental feature of an Indeterminacy set is that it generalizes classical, fuzzy, and intuitionistic fuzzy sets by explicitly incorporating indeterminacy. Example 2.3. Let S = { α , β } . Define An Indeterminacy set A as: A = { ( α , 0.7 , 0.2 , 0.1 ) , ( β , 0.4 , 0.3 , 0.6 ) } . Here, element a belongs to A with truth degree 0.7, indeterminacy 0.2, and falsity 0.1. Definition 2.4. 12 For ( G , ⋅ ) , We have that it is a group. An Indeterminacy group is defined as: ( G ∪ I ) = { α + βI : α , β ∈ G } , where I is an indeterminate element with I ² = I . The group operations are extended naturally from G . Example 2.5. For G = ( Z 3 , + ) the additive group of integers modulo 3. So, Indeterminacy group is: ( G ∪ I ) = { 0 , 1 , 2 , 0 + I , 1 + I , 2 + I } . For example, ( 1 + I ) + ( 2 + I ) = ( 1 + 2 ) + ( I + I ) = 0 + I . Definition 2.6. 8 For R to be a ring. R I = { α + βI : α , β ∈ R } is referred to as Indeterminacy ring generated by R and I . Remark 2.7. The indeterminate element I satisfies the condition I ² = I , which is essential in defining Indeterminacy rings. Definition 2.8. 3 Let R be a ring. The Indeterminacy ring R I is a ring generated by R and I . Remark 2.9. The angle bracket notation R I is sometimes used to emphasize the closure under ring operations. Example 2.10. 6 We denote by Z the ring of integers, Z I = { α + βI : α , β ∈ Z } . This is a ring termed the Indeterminacy ring of integers. Also, Z ⊊ Z I . Remark 2.11. The enlargement from Z to Z I highlights how Indeterminacy extensions generalize classical rings. Example 2.12. 4 We denote by Q the ring of rationales. Q I = { λ + βI : λ , β ∈ Q } . This is the Indeterminacy ring of rationales. Definition 2.13. 13 We denote by R a ring. A subset J I ⊆ R I is referred to as Indeterminacy ideal of the Indeterminacy ring R I if for all r ∈ R and j ∈ J , we have: rj ∈ J , jr ∈ J , and ( j + I ) r , r ( j + I ) ∈ J I . Remark 2.14. The presence of the indeterminate I ensures that classical ideals extend naturally into the Indeterminacy framework. Example 2.15. Let R = Z 6 and J = { 0 , 2 , 4 } . Then J is an ideal in R . The Indeterminacy ideal is: J I = { 0 , 2 , 4 , 0 + I , 2 + I , 4 + I } ⊆ Z 6 I . Definition 2.16. 14 An Indeterminacy field is An Indeterminacy algebraic structure ( F I , + , ⋅ ) where F is a classical field and I is the Indeterminacy indeterminate with I ² = I . It satisfies all field axioms extended with the Indeterminacy component. Example 2.17. Let F = Q , the field of rational numbers. Then the Indeterminacy field is: F I = { α + βI : α , β ∈ Q } . For example, ( 1 + I ) ⋅ ( 2 + I ) = 2 + 3 I + I ² = 2 + 3 I + I = 2 + 4 I . Remark 2.18. Although Q is a field, Q I is not a field since I ² = I and I has no multiplicative inverse. Still, it is sometimes loosely designated the Indeterminacy field of rationales. Example 2.19. 5 Let R be fixed as the ring of real numbers. R I = { λ + sI : λ , s ∈ R } . This is the Indeterminacy ring of real’s. Remark 2.20. Similarly to the rationals, it is only a ring and not a true field, but in literature it is sometimes termed the Indeterminacy field of real’s. Example 2.21. 8 C I = { z + wI : z , w ∈ C . This is the Indeterminacy ring of complex numbers. Remark 2.22. Even though C is algebraically closed and a field, its Indeterminacy extension C I is not a field because of the special Indeterminacy element I. Definition 2.23. Let R I be An Indeterminacy ring. It is said to be comm. if ∀ αI , βI ∈ R I , ( αI ) ( βI ) = ( βI ) ( αI ) . If in addition ∃ I ∈ R I ∋ 1 · λ = λ · 1 = λ , ∀ λ ∈ R I , then R I is termed a comm. Indeterminacy ring with unity. Remarks 2.24. 1. 7 Unity here generalizes the multiplicative identity of the base ring R . 2. 3 Again, although Q is a field, Q I is only a ring since I lack an inverse. 3. It is not a field, but in many Indeterminacy studies it is referred to as the Indeterminacy field of complex numbers. 3. Results In this part, we present some new results about important ring in abstract algebra and more properties have been studied. But before that we, need some definitions and examples in order to study this ring in Indeterminacy theory. Definition 3.1. 10 Any ring R is termed Boolean if α 2 = α , ∀ α ∈ R ). Definition 3.2. For a B-Ring R , We say R I is Indeterminacy B-Ring if for every element in R I αI ∈ R I , then ( αI ) 2 = α 2 I 2 = αI . Example 3.3. An Indeterminacy ring ( Z 2 I , + 2 , . 2 ) is An Indeterminacy B-Ring. Example 3.4. An Indeterminacy ring ( P ( X ) , ∆ , ∩ ) , Where P ( X ) = { AI : AI ⊆ X I } is An Indeterminacy B-Ring Since ( P ( X ) , ∆ , ∩ ) is An Indeterminacy ring with identity and ∀ AI ∈ P ( X ) ⟹ A 2 I 2 = AI ∩ AI = AI . Example 3.5. An Indeterminacy ring ( Z 3 I , + 3 , . 3 ) is not Indeterminacy B-Ring. Since ∃ 2 I ¯ ∈ Z 3 I is not idempotent element, s.t 2 2 I 2 = 2 I . 3 2 I = I ≠ 2 I . Example 3.6. For ( R I , + , ∙ ) comm. Indeterminacy ring with unity R I = { ψ ; ψ : X I → Z 2 I } and ∀ αI ∈ X I we have: ( ψ + ϕ ) ( αI ) = ψ ( αI ) + 2 ϕ ( αI ) ( ψϕ ) ( αI ) = ψ ( αI ) ∙ 2 ϕ ( αI ) So, ( R I , + , ∙ ) be a comm. Indeterminacy ring with unity. It achieves the following and ∀ ψ ∈ R I either ψ ( αI ) = 0 or ψ ( αI ) = 1 . Then ψ 2 = ψ . If ψ ( αI ) = 0 . Then, ψ ( αI ) 2 = ψ ( αI ) ∙ 2 ψ ( αI ) = 0 ∙ 2 0 = 0 Or ψ ( αI ) = 1 Hence, ψ ( αI ) 2 = ψ ( αI ) ∙ 2 ψ ( αI ) = I ∙ 2 I = I And ψ 2 = ψ Thus, ( R I , + , ∙ ) Indeterminacy B-Ring. Theorem 3.7. Consider R I as An Indeterminacy B-Ring. Then ( − αI ) = αI , ∀ αI ∈ R I . Proof: We prove that if R I is a ring, so ∀ αI , βI ∈ R I , ( − αI ) ( − βI ) = αβI and ( − αI ) 2 = αI , ( − αI ) ( − αI ) = ( αI ) 2 . From the definition of An Indeterminacy B-Ring, ( αI ) 2 = α 2 I 2 = αI ⟹ ( αI ) 2 = αI … . . ∗ . Thus, ( − αI ) 2 = ( − αI ) . But ( − αI ) 2 = ( αI ) 2 . So, ( αI ) 2 = ( − αI ) . from * we get αI = ( − αI ) and this required. Theorem 3.8. Every Indeterminacy B-Ring ( R I , + , . ) with the characteristic 2 has the property 2 αI = 0 I . Proof: Let αI ∈ R I , and since R I is An Indeterminacy B-ring Therefore, αI + αI ∈ R I So, αI + αI = ( αI + αI ) 2 ( R I is a B-Ring) Hence, αI + αI = α 2 I 2 + 2 α 2 I 2 + α 2 I 2 Therefore, αI + αI = αI + 2 αI + αI ( αI ∈ R I ⟹ α 2 I 2 = αI ) Then, 0 = 2 αI ⟹ 2 αI = 0 , ∀ αI ∈ R I Thus, h ( R ) = 2 . Theorem 3.9. Consider R I as An Indeterminacy B-Ring. Then R I is comm. under ( · ) . Proof: Assume that I , βI ∈ R I . We need to show that αβI = βαI s . t ( αI ) ( βI ) = αβ I 2 = αβI ⟹ ( αI + βI ) = ( αI + βI ) 2 (since αI , βI ∈ R I and is Indeterminacy B-Ring) ⟹ ( αI + βI ) = ( αI + βI ) ( αI + βI ) ⟹ ( αI + βI ) = ( αI ) 2 + αIβI + βIαI + ( βI ) 2 But R I An Indeterminacy B-Ring we have ( αI ) 2 = αI and ( βI ) 2 = βI ⟹ ( αI + βI ) = αI + αIβI + βIαI + βI ⟹ 0 = αIβI + βIαI ⟹ αIβI = − αIβI but αI = − αI from Theorem 3.7 Hence αIβI = αIβI as required. Theorem 3.10. If R I is An Indeterminacy ring with identity. Then, every Indeterminacy maximal ideal is Indeterminacy prime ideal. Proof: Assume that R I Indeterminacy ring with identity and ( P I , + , ∙ ) An Indeterminacy maximal ideal in R I . To verify that ( P I , + , ∙ ) is Indeterminacy prime ideal. Let αI , βI ∈ R I , ∋ αI ∙ βI ∈ P I and let αI ∉ P I ( P I , + , ∙ ) An Indeterminacy maximal ideal in R I , and αI ∉ P I . Then PI + ( αI ) = R I . Hence I ∈ R I ⟹ I ∈ PI + ( αI ) ⟹ I = δI + λIαI , λI ∈ R I , δI ∈ P I and I 2 = I , I = δI + λαI } ∗ βI βI ∙ I = δI ∗ βI + λIαI ∗ βI , βI = δβI + λαβI . Therefore δI ∈ P I , βI ∈ R I ⟹ δβI ∈ P I . Hence λI ∈ R I , αI ∙ βI ∈ P I ⟹ λαβI ∈ P I . Then δβI + λαβI ∈ P I . So βI ∈ P I . Thus ( P I , + , ∙ ) is Indeterminacy prime ideal . Theorem 3.11. Let R I be An Indeterminacy B-Ring and S I be An Indeterminacy ideal in that Indeterminacy ring. Then P is Indeterminacy prime ideal iff it is Indeterminacy maximal ideal. Proof: ⟹ Let S I Indeterminacy prime ideal. We need to prove that S I is Indeterminacy maximal ideal. Take an P I Indeterminacy ideal in R I . s.t ( S ⊂ P ⊆ R) S I ⊂ P I ⊆ R I , To show that P I = R I . Hence S I ⊂ P I ⟹ ∃ αI ∈ P I , αI ∉ S I . Then αI ∈ P I ⟹ αI ∈ R I since P I ⊆ R I . But R I is Indeterminacy B-Ring, then ( αI ) 2 = αI αI ( I − αI ) = 0 ∈ S I . So αI ∉ S I , S I Indeterminacy prime ideal. Therefore, ( I − αI ) ∈ S I , Hence S I ⊂ P I , And so, ( I − αI ) ∈ P I , . Also, ( I − αI ) + αI ∈ P I . Then, I ∈ P I , So P I = R I . Thus, S I is Indeterminacy maximal ideal. ⟸ Let S I be An Indeterminacy maximal ideal. To prove S I is An Indeterminacy prime ideal. Hence R I is An Indeterminacy B-Ring. Also, R I abelian Indeterminacy ring with identity and by Theorem 3.11 (if R I Indeterminacy ring with identity. Then, every Indeterminacy maximal ideal is Indeterminacy prime ideal). Thus S I Indeterminacy prime ideal. Remark 3.12. (1) Let ( R I , + , ∙ ) be a comm. Indeterminacy ring with ( 1 I ) , The S I ⊆ R I such that a) S I = ( αI ∈ R I | ( αI ) 2 = αI ) , b) αI + βI = αI + βI − 2 αβI , αI ∗ βI = αβI , ∀ αI , βI ∈ R I . Then, ( S I , + , ∙ ) is An Indeterminacy B-Ring. S I = ( αI ∈ R I | ( αI ) 2 = αI ) , ∀ αI , βI ∈ R I αI + βI = αI + βI − 2 αβI , αI ∗ βI = αβI . (2) Multiplication: If αI , βI ∈ S I , then ( αβI ) ² = ( αI ) ² ( βI ) ² = αβI , so αβI ∈ S I . (3) Addition: If αI , βI ∈ S I , then αI + βI = αI + βI − 2 αβI . Also, ( αI + βI − 2 αβI ) ² = ( αI + βI − 2 αβI ) ( αI + βI − 2 αβI ) = ( αI + βI ) ( αI + βI ) − 2 αβI ( αI + βI ) − 2 αβI ( αI + βI ) + ( 2 αβI ) ( 2 αβI ) = ( αI ) 2 + 2 αβI + ( βI ) 2 − 2 ( αI ) 2 βI − 2 αI ( βI ) 2 − 2 ( αI ) 2 βI − 2 αI ( βI ) 2 + 4 ( αI ) 2 ( βI ) 2 = αI + 2 αβI + βI − 2 αβI − 2 αβI − 2 αβI − 2 αβI + 4 αβI = αI + βI − 2 αβI , so αI + βI ∈ S I , also ( S I , + ) is an abelian group. (4) Comm.: αI + βI = βI + αI Associativity: holds by expansion. Identity of ( S I , + ) is 0 s.t I + 0 = 0 + αI = αI . Inverses in ( S I , + ) , αI + α − 1 I = 0 , so every element is its own inverse. Thus ( S I , + ) is an abelian group. (5) Distributive: For αI , βI , γI ∈ S I : αI ( βI + γI ) = αβI + αγI − 2 αβγI = ( αβI ) + ( αγI ) . So distributivity holds. (6) Boolean property: For every αI ∈ S I : each element is idempotent under multiplication, and ( S I , + , ∙ ) satisfies all ring axioms and every element is idempotent under multiplication. Hence ( S I , + , ∙ ) is a B-Ring. Corollary 3.13. If ( R I , + , . ) is An Indeterminacy ring with identity I . Then, ( R I / P I , + , . ) is also Indeterminacy ring with identity I + δI . Proof: Since R I is An Indeterminacy ring with identity I , Then + δI ∈ R I / P I . Hence I + δ I is an identity element of R I / P I with respect multiplication, Since ∀ αI + δI ∈ R I / P I . Hence ( αI + δ I ) . ( I + δ I ) = ( αI . I ) + pI = αI + δ I and ( I + δ I ) . ( αI + δ I ) = ( I . αI ) + δ I = ( αI + δ I ) . Therefore, R I / P I is Indeterminacy ring with identity element + δI . Theorem 3.14. For ( R I , + , . ) , An Indeterminacy B-Ring. Then, ( R I P I , + , . ) is also Indeterminacy B-Ring. Theorem 3.15. Consider ( R I , + , · ) as An Indeterminacy B-Ring. For any αI ≠ 0 , βI ≠ 0 , γI ≠ 0 ∈ R I , Then ( αI + βI ) ( βI + γI ) ( γI + αI ) = 0 . Proof: In An Indeterminacy B-Ring, every element is idempotent, that is ( αI ) ² = αI for all αI ∈ R I . 7 From idempotency, one derives that the ring has characteristic 2 . Indeed, ( αI + αI ) ² = αI + αI implies 2 αI = 0 , hence αI + αI = 0 for all αI ∈ R I . 9 Since ( αI ) ² = αI and ( βI ) ² = βI , we have αβI + βαI = 0 . In characteristic 2 , this simplifies to αβI = βαI . An Indeterminacy B-Ring is comm. So, ( αI + βI ) ² = ( αI ) ² + αβI + βαI + ( βI ) ² = αI + βI . 5 Multiplying this ( αI + βI ) ( βI + γI ) ( γI + αI ) , ∀ αI , βI , γI ∈ ( R ∪ I ) s . t αI , βI and γI non-zero element. Now ( αI + βI ) ( βI + γI ) = αβI + αγI + ( βI ) 2 + βγI = αβI + αγI + βI + βγI , Now ( αI + βI ) ( βI + γI ) ( γI + αI ) = ( αβI + αγI + βI + βγI ) ( γI + αI ) = ( αβγI + αβγI ) + ( αβI + αβI ) + ( αγI + αγI ) + ( βγI + βγI ) = 2 αβγI + 2 αβI + 2 αγI + 2 βγI = 0 + 0 + 0 + 0 . Then ( αI + βI ) ( βI + γI ) ( γI + αI ) = 0 . 6 Corollary 3.16. Let ( P I , + , ∙ ) be a proper ideal in the Indeterminacy B-Ring ( R I , + , · ) . Then P I is maximal iff ( R I / P I , + , · ) ≅ ( Z 2 I , + 2 , · 2 ) . Proof: Since ( R I , + , · ) is an Indeterminacy B-Ring, the quotient Indeterminacy ring ( R I / P I , + , · ) is also Indeterminacy Boolean. Moreover, as R I is a comm. Indeterminacy ring with identity, R I / P I inherits these properties and remains a comm. Indeterminacy ring with identity. For any element αI + P I ∈ R I / P I , we have: ( αI + P I ) ² = ( αI + P I ) ( αI + P I ) = αI ² + P I = αI + P I . Hence, R I / P I is an Indeterminacy B-Ring. It is well known that an ideal P I is maximal in R I iff R I / P I is an Indeterminacy field. Corollary 3.17. Every Indeterminacy B-Ring ( R I , + , · ) is semisimple, that is, rαd ( R I ) = { 0 } . Proof: Assume that ( R I , + , · ) be an Indeterminacy B-Ring. We aim to prove that R I is semisimple, i.e., rαd ( R I ) = { 0 } . Let, for contradiction, that rαd ( R I ) ≠ { 0 } . Then there exists a nonzero element αI ∈ rαd ( R I ) . From the auxiliary lemma, there exists an Indeterminacy ring homomorphism ψ : R I → Z 2 I such that ψ ( αI ) = I . Consequently, ker ( ψ ) is a proper ideal of R I , and hence there exists a maximal ideal P I in R I such that ker ( ψ ) ⊆ P I . Since 1 I − αI ∈ ker ( ψ ) ⊆ P I and αI ∈ P I (because αI ∈ rαd ( R I ) = P I -maximal), we get 1 I = αI + ( 1 I − αI ) ∈ P I , which implies that P I = R I , a contradiction. Therefore, rαd ( R I ) = { 0 } , and R I is semisimple. Theorem 3.18. Every Indeterminacy B-Ring is isomorphic to a direct product of copies of Z 2 I . Formally, R I ≅ ∏ { i ∈ I } Z 2 I . Proof: Every Indeterminacy B-Ring can be viewed as an Indeterminacy ring of functions from some index set P I to Z 2 I . This representation arises because each element of R I corresponds to unique Indeterminacy boolean combination of projections onto Z 2 I . Proposition 3.19. Every ideal in an Indeterminacy B-Ring is Indeterminacy radical ideal. Proof: Assume that P I an ideal of R I , and suppose αI ∈ P I , meaning αⁿ ∈ P I for some n ≥ 1 . But in an Indeterminacy B-Ring, ( αI ) ⁿ = αI , so αI ∈ P I . Hence, P I = P I . Theorem 3.20. Every Indeterminacy B-Ring is reduced (contains no nonzero nilpotent elements) and therefore semi-simple. Proof: If αI ∈ rαd ( R I ) , then a is nilpotent. But in an Indeterminacy B-Ring, ( αI ) 2 = αI , implying αI = 0 or 1 I . Since 1 I ∉ rαd ( R I ) , it follows that αI = 0 . Thus, rαd ( R I ) = 0 , and R I is semisimple. Proposition 3.21. For each element αI in an Indeterminacy B-Ring R I , there exists an onto Indeterminacy ring homomorphism ψ : R I → Z 2 I such that ψ ( αI ) = 1 I . This shows that Indeterminacy B-Ring possess many surjective homomorphisms to Z 2 I , allowing R I to decompose as a direct product of copies of Z 2 I . Corollary 3.22. Up to isomorphism, there exists only one Boolean field, namely Z 2 I . Proposition 3.23. An Indeterminacy B-Ring ( R I , + , · ) is an Indeterminacy field iff ( R I / P I , + , · ) ≅ ( Z 2 I , + 2 , · 2 ) . Proof: Assume that ( R I / P I , + , · ) is an Indeterminacy B-Field. For any αI ∈ R I , the following holds: αI = αI · 1 I = αI ( αI · ( αI ) − 1 ) = ( αI ) ² · ( αI ) − 1 = αI · ( αI ) − 1 = 1 I . Thus, R I = { 0 , 1 I } , and consequently R I ≅ Z 2 I . Therefore, P I is maximal in R I iff R I / P I is an Indeterminacy field, which occurs precisely when R I / P I ≅ Z 2 I . Conclusion Boolean ring is one of the important rings in abstract algebra. All results and properties of Boolean ring are presented in Indeterminacy theory. We proved in ( Theorem 3.7 ) the relation between Indeterminacy Boolean ring and the property ( − αI = αI .). Also, we proved that, every Indeterminacy Boolean ring ( R I , + , . ) with the characteristic 2 has the property 2 αI = 0 I . On the other hand, if R I is An Indeterminacy ring with identity so, every Indeterminacy maximal ideal is Indeterminacy prime ideal. In Corollary 3.14, we say if ( R I , + , . ) is An Indeterminacy ring with identity I . Then, ( R I / P I , + , . ) is also Indeterminacy ring with identity I + δI . Finally, more results in this article have been presented. Discussion This paper does not include a discussion section. Ethical considerations This article does not involve human participants or animal subjects. Data availability No datasets were generated or analyzed during the current study. Reporting guidelines All relevant research and reporting guidelines were appropriately followed. Acknowledgements The author would like to thank the reviewers for their valuable and constructive comments that helped improve the quality of the article. References 1. Smarandache F: (T, I, F)-Neutrosophic structures and their applications. Neutrosophic Sets and Systems. 2015; 8 : 15–28. 2. Agboola AA, Akinola AD, Oyebola OY: Neutrosophic Rings I. International J.Math. Combin. 2011; 4 : 1–14. 3. Chalapathi T, Madhavi L: Neutrosophic Boolean rings. Neutrosophic Sets and Systems. 2020; 33 : 59–66. 4. Al-Hamido RK: A New Neutrosophic Algebraic Structures. Journal of Computational and Cognitive Engineering. 2022; 2 : 150–154. 5. Agboola AA, Adeleke EO, Akinleye AA: Neutrosophic Rings II. International J.Math. Combin. 2012; 2 : 1–8. 6. Hussain A, Shabir M: Algebraic Structures of Neutrosophic Soft Sets. Neutrosophic Sets and Systems. 2015; 7 (1). 7. Al-Odhari AM: Axiomatic of Neutrosophic Groups. Sana’a University Journal of Applied Sciences and Technology. 2024; 2 (2): 205–214. 8. Jacobe A, Vilela M: Introduction to neutrosophic B-algebras and related structures. European Journal of Pure and Applied Mathematics. 2021; 14 (3): 895–904. 9. Wang H, Zhang YQ, Smarandache F: Recent developments in neutrosophic algebraic computation and applications. Symmetry. 2019; 11 (1): 1–15. 10. Al-Odhari AM: Basic Introduction of Neutrosophic Set Theory. Plithogenic Logic and Computation. 2024; 2 : 20–28. 11. Smarandache F: n-Refined Neutrosophic Rings. International Journal of Neutrosophic Science. 2020; 5 (2): 83–90. 12. Şahin R: Neutrosophic group and its applications to decision making problems. J. Intell. Fuzzy Syst. 2014; 27 (5): 2417–2430. 13. Broumi S, Smarandache F: Neutrosophic ideal theory and its applications. Neutrosophic Sets and Systems. 2013; 1 : 50–59. 14. Ali M, Smarandache F, Hassan N: Neutrosophic fields and their applications. International Journal of Neutrosophic Science. 2017; 1 (1): 10–20. 15. Kareem FF, Abed MM: Generalizations of fuzzy k-ideals in a KU-algebra with semigroup. J. Phys. Conf. Ser. 2021; 1879 : 022108. 16. Abed MM: On indeterminacy (neutrosophic) of hollow modules. Iraqi Journal of Science. 2022; 2650–2655. 17. Abed MM, Talak AF, Hameed FN: An approach to singular modules by indeterminacy concept. International Journal of Neutrosophic Science (IJNS). 2023; 21 (4): 30–35. Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 06 Feb 2026 ADD YOUR COMMENT Comment Author details Author details 1 Mathematic, University of Anbar, Ramadi, Al Anbar Governorate, Iraq Yousif A. Ahmed Roles: Writing – Review & Editing Majid Mohammed Abed Roles: Writing – Review & Editing Competing interests No competing interests were disclosed. Grant information The author(s) declared that no grants were involved in supporting this work. Article Versions (1) version 1 Published: 06 Feb 2026, 15:205 https://doi.org/10.12688/f1000research.172934.1 Copyright © 2026 Ahmed YA and Mohammed Abed M. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The author(s) is/are employees of the US Government and therefore domestic copyright protection in USA does not apply to this work. The work may be protected under the copyright laws of other jurisdictions when used in those jurisdictions. Download Export To Sciwheel Bibtex EndNote ProCite Ref. 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Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 1 VERSION 1 PUBLISHED 06 Feb 2026 Views 0 Cite How to cite this report: Gokavarapu C. Reviewer Report For: Indeterminacy of Boolean Ring [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :205 ( https://doi.org/10.5256/f1000research.190701.r473985 ) The direct URL for this report is: https://f1000research.com/articles/15-205/v1#referee-response-473985 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 06 May 2026 Chandrasekhar Gokavarapu , Acharya Nagarjuna University, Nagarjuna Nagar, Andhra Pradesh, India Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.190701.r473985 Detailed Report 1. Presentation and Literature Review The introduction successfully contextualizes the work within fuzzy theory and Smarandache’s neutrosophic framework. However, there is a distinct "organizational gap" between the literature review and the technical proofs. ... Continue reading READ ALL Detailed Report 1. Presentation and Literature Review The introduction successfully contextualizes the work within fuzzy theory and Smarandache’s neutrosophic framework. However, there is a distinct "organizational gap" between the literature review and the technical proofs. Criticism: The transitions between the historical context and the new mathematical developments are abrupt. Recommendation: Strengthen the narrative by explicitly stating how the cited literature necessitates the specific theorems presented in Section 3. 2. Technical Soundness and Methodology While the mathematical logic is sound and the theorems are relevant, the Methods/Preliminary section (Section 2) suffers from a lack of cohesion. Criticism: Definitions and examples often appear disconnected, making it difficult for the reader to grasp the logical progression. Recommendation: Group related definitions (e.g., combining the formal definition of an Indeterminacy B-ring with its illustrative example) to improve clarity. Introductory paragraphs should be added to each subsection to explain the utility of each concept. 3. Language and Formal Aspects The most significant barrier to the article's impact is its current editorial state. Grammar & Punctuation: There are frequent missing commas and periods, and inconsistent capitalization (e.g., "An indeterminacy" vs. "an indeterminacy"). Mathematical Style: The text frequently embeds symbolic quantifiers (like $\forall$ or $\exists$) directly into prose sentences. Recommendation: Replace symbolic quantifiers with words (e.g., "for all" or "there exists") when they appear in the middle of a sentence to improve readability. A comprehensive professional proofreading is essential. 4. Conclusions The conclusions are logically derived from the results, but they remain somewhat brief. Recommendation: Expand the conclusion to discuss specific potential applications in fuzzy systems or neutrosophic logic, providing a "roadmap" for future research. Required Amendments for Scientific Soundness To move this article from "Approved with Reservations" to full approval, the following points must be addressed: Reorganize Section 2: Create a clearer logical flow by adding transitions and grouping interdependent definitions. Standardize Notation: Ensure the ring notation is consistent throughout the manuscript, particularly the transition between $R_L$ and $R_I$. Language Correction: Fix grammatical errors, standardize capitalization, and ensure punctuation is mathematically and linguistically correct. Contextualize Definitions: Provide a brief explanation for why a specific definition (e.g., Indeterminacy Field) is being introduced and how it serves the subsequent theorems. Is the work clearly and accurately presented and does it cite the current literature? Partly Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? No source data required Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: ALGEBRA, SYMMETRY, NONLINEAR DYNAMICS I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Gokavarapu C. Reviewer Report For: Indeterminacy of Boolean Ring [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :205 ( https://doi.org/10.5256/f1000research.190701.r473985 ) The direct URL for this report is: https://f1000research.com/articles/15-205/v1#referee-response-473985 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Higuera Rincon SD and Rubiano Suárez AA. Reviewer Report For: Indeterminacy of Boolean Ring [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :205 ( https://doi.org/10.5256/f1000research.190701.r456955 ) The direct URL for this report is: https://f1000research.com/articles/15-205/v1#referee-response-456955 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 05 Mar 2026 Sebastian David Higuera Rincon , Universidad Antonio Narino (Ringgold ID: 27967), Bogotá, Bogotá, Colombia Andrés Alejandro Rubiano Suárez , Escuela Colombiana De Carreras Industriales (Ringgold ID: 268828), Bogotá, Bogotá, Colombia Approved with Reservations VIEWS 0 https://doi.org/10.5256/f1000research.190701.r456955 The submitted article addresses a relevant topic in its research area and presents an interesting and potentially valuable contribution to the literature. Below, I provide my evaluation regarding clarity of presentation, methodological soundness, and the connection between results and conclusions. ... Continue reading READ ALL The submitted article addresses a relevant topic in its research area and presents an interesting and potentially valuable contribution to the literature. Below, I provide my evaluation regarding clarity of presentation, methodological soundness, and the connection between results and conclusions. 1. General Presentation and Literature Review The introduction of the article is clear and well written. It explains the current state of the topic in the literature and clearly defines the problem that the paper aims to study. The objectives of the work are presented in a precise way, and the potential contribution of the article is understandable. However, it would be helpful to strengthen the connection between the literature review and the technical developments that appear later in the paper. Adding clearer transitions could help readers better understand the importance of the main results from the beginning. 2. Methods and Preliminary Section The Methods (or preliminary) section includes many definitions, examples, and remarks that seem necessary for the development of the paper. However, this section is not always clearly organized. The information sometimes appears disconnected, and it is difficult to see a clear logical structure. Although the preliminary material is important, I suggest the following improvements: Provide more explanation about the context of each definition or concept. Explain more clearly why each concept is important in the existing literature. Indicate how each definition or result will be used later in the Results section. Add short introductory paragraphs or transitions to guide the reader through the section. In addition, some definitions and examples could be grouped together to avoid repetition and improve clarity. For example, the definition of Indeterminacy B-ring could be presented in a more compact way, combining the formal definition and an illustrative example in the same place. These changes would improve the clarity and coherence of the paper. 3. Results and Conclusions The Results section is generally well developed. The theorems presented are relevant and meet the expectations created in the introduction. There is a good connection between the objectives of the paper and the results obtained. The conclusions are consistent with the results. However, the final section could be strengthened by emphasizing more clearly the importance of the results, their possible applications, and potential directions for future research. 4. Writing and Formal Aspects One of the main aspects that needs improvement is the writing quality. The article contains several grammatical and punctuation errors that should be corrected before indexing. In particular: Some commas and final periods are missing. The ring notation should be carefully revised for consistency. There are incorrect expressions in English, such as writing “An indeterminacy” instead of “an indeterminacy”. In several places, symbolic quantifiers are used in the middle of sentences. It would be better to write them in words (for example, “for all x” or “there exists an x such that…”) to improve readability and style. A careful language revision by a native English speaker or a professional proofreader is strongly recommended. 5. Overall Evaluation In conclusion, this is an excellent work from a mathematical and conceptual point of view. The results are relevant and aligned with the objectives stated in the introduction. However, the paper needs important improvements in the organization of the preliminary section and especially in the writing and presentation of the material. The following points must be addressed to ensure that the article is scientifically sound and clearly presented: Reorganize and clarify the preliminary section. Provide more context for the definitions and concepts introduced. Carefully revise grammar, punctuation, and language. Correct and standardize the mathematical notation. After these revisions, the article would have strong potential to become a solid and well-presented contribution to the field. Is the work clearly and accurately presented and does it cite the current literature? Partly Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? I cannot comment. A qualified statistician is required. Are all the source data underlying the results available to ensure full reproducibility? No source data required Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: My research areas are commutative and noncommutative algebra of polynomial type, module theory, and category theory. We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however we have significant reservations, as outlined above. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Higuera Rincon SD and Rubiano Suárez AA. Reviewer Report For: Indeterminacy of Boolean Ring [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :205 ( https://doi.org/10.5256/f1000research.190701.r456955 ) The direct URL for this report is: https://f1000research.com/articles/15-205/v1#referee-response-456955 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 06 Feb 2026 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 2 Version 1 06 Feb 26 read read Sebastian David Higuera Rincon , Universidad Antonio Narino (Ringgold ID: 27967), Bogotá, Colombia Andrés Alejandro Rubiano Suárez , Escuela Colombiana De Carreras Industriales (Ringgold ID: 268828), Bogotá, Colombia Chandrasekhar Gokavarapu , Acharya Nagarjuna University, Nagarjuna Nagar, India Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Gokavarapu C. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 06 May 2026 | for Version 1 Chandrasekhar Gokavarapu , Acharya Nagarjuna University, Nagarjuna Nagar, Andhra Pradesh, India 0 Views copyright © 2026 Gokavarapu C. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Detailed Report 1. Presentation and Literature Review The introduction successfully contextualizes the work within fuzzy theory and Smarandache’s neutrosophic framework. However, there is a distinct "organizational gap" between the literature review and the technical proofs. Criticism: The transitions between the historical context and the new mathematical developments are abrupt. Recommendation: Strengthen the narrative by explicitly stating how the cited literature necessitates the specific theorems presented in Section 3. 2. Technical Soundness and Methodology While the mathematical logic is sound and the theorems are relevant, the Methods/Preliminary section (Section 2) suffers from a lack of cohesion. Criticism: Definitions and examples often appear disconnected, making it difficult for the reader to grasp the logical progression. Recommendation: Group related definitions (e.g., combining the formal definition of an Indeterminacy B-ring with its illustrative example) to improve clarity. Introductory paragraphs should be added to each subsection to explain the utility of each concept. 3. Language and Formal Aspects The most significant barrier to the article's impact is its current editorial state. Grammar & Punctuation: There are frequent missing commas and periods, and inconsistent capitalization (e.g., "An indeterminacy" vs. "an indeterminacy"). Mathematical Style: The text frequently embeds symbolic quantifiers (like $\forall$ or $\exists$) directly into prose sentences. Recommendation: Replace symbolic quantifiers with words (e.g., "for all" or "there exists") when they appear in the middle of a sentence to improve readability. A comprehensive professional proofreading is essential. 4. Conclusions The conclusions are logically derived from the results, but they remain somewhat brief. Recommendation: Expand the conclusion to discuss specific potential applications in fuzzy systems or neutrosophic logic, providing a "roadmap" for future research. Required Amendments for Scientific Soundness To move this article from "Approved with Reservations" to full approval, the following points must be addressed: Reorganize Section 2: Create a clearer logical flow by adding transitions and grouping interdependent definitions. Standardize Notation: Ensure the ring notation is consistent throughout the manuscript, particularly the transition between $R_L$ and $R_I$. Language Correction: Fix grammatical errors, standardize capitalization, and ensure punctuation is mathematically and linguistically correct. Contextualize Definitions: Provide a brief explanation for why a specific definition (e.g., Indeterminacy Field) is being introduced and how it serves the subsequent theorems. Is the work clearly and accurately presented and does it cite the current literature? Partly Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? No source data required Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise ALGEBRA, SYMMETRY, NONLINEAR DYNAMICS I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above. reply Respond to this report Responses (0) Gokavarapu C. Peer Review Report For: Indeterminacy of Boolean Ring [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :205 ( https://doi.org/10.5256/f1000research.190701.r473985) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/15-205/v1#referee-response-473985 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Higuera Rincon S et al. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 05 Mar 2026 | for Version 1 Sebastian David Higuera Rincon , Universidad Antonio Narino (Ringgold ID: 27967), Bogotá, Bogotá, Colombia Andrés Alejandro Rubiano Suárez , Escuela Colombiana De Carreras Industriales (Ringgold ID: 268828), Bogotá, Bogotá, Colombia 0 Views copyright © 2026 Higuera Rincon S et al. This is an open access peer review report distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. format_quote Cite this report speaker_notes Responses (0) Approved With Reservations info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The submitted article addresses a relevant topic in its research area and presents an interesting and potentially valuable contribution to the literature. Below, I provide my evaluation regarding clarity of presentation, methodological soundness, and the connection between results and conclusions. 1. General Presentation and Literature Review The introduction of the article is clear and well written. It explains the current state of the topic in the literature and clearly defines the problem that the paper aims to study. The objectives of the work are presented in a precise way, and the potential contribution of the article is understandable. However, it would be helpful to strengthen the connection between the literature review and the technical developments that appear later in the paper. Adding clearer transitions could help readers better understand the importance of the main results from the beginning. 2. Methods and Preliminary Section The Methods (or preliminary) section includes many definitions, examples, and remarks that seem necessary for the development of the paper. However, this section is not always clearly organized. The information sometimes appears disconnected, and it is difficult to see a clear logical structure. Although the preliminary material is important, I suggest the following improvements: Provide more explanation about the context of each definition or concept. Explain more clearly why each concept is important in the existing literature. Indicate how each definition or result will be used later in the Results section. Add short introductory paragraphs or transitions to guide the reader through the section. In addition, some definitions and examples could be grouped together to avoid repetition and improve clarity. For example, the definition of Indeterminacy B-ring could be presented in a more compact way, combining the formal definition and an illustrative example in the same place. These changes would improve the clarity and coherence of the paper. 3. Results and Conclusions The Results section is generally well developed. The theorems presented are relevant and meet the expectations created in the introduction. There is a good connection between the objectives of the paper and the results obtained. The conclusions are consistent with the results. However, the final section could be strengthened by emphasizing more clearly the importance of the results, their possible applications, and potential directions for future research. 4. Writing and Formal Aspects One of the main aspects that needs improvement is the writing quality. The article contains several grammatical and punctuation errors that should be corrected before indexing. In particular: Some commas and final periods are missing. The ring notation should be carefully revised for consistency. There are incorrect expressions in English, such as writing “An indeterminacy” instead of “an indeterminacy”. In several places, symbolic quantifiers are used in the middle of sentences. It would be better to write them in words (for example, “for all x” or “there exists an x such that…”) to improve readability and style. A careful language revision by a native English speaker or a professional proofreader is strongly recommended. 5. Overall Evaluation In conclusion, this is an excellent work from a mathematical and conceptual point of view. The results are relevant and aligned with the objectives stated in the introduction. However, the paper needs important improvements in the organization of the preliminary section and especially in the writing and presentation of the material. The following points must be addressed to ensure that the article is scientifically sound and clearly presented: Reorganize and clarify the preliminary section. Provide more context for the definitions and concepts introduced. Carefully revise grammar, punctuation, and language. Correct and standardize the mathematical notation. After these revisions, the article would have strong potential to become a solid and well-presented contribution to the field. Is the work clearly and accurately presented and does it cite the current literature? Partly Is the study design appropriate and is the work technically sound? Partly Are sufficient details of methods and analysis provided to allow replication by others? Partly If applicable, is the statistical analysis and its interpretation appropriate? I cannot comment. A qualified statistician is required. Are all the source data underlying the results available to ensure full reproducibility? No source data required Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise My research areas are commutative and noncommutative algebra of polynomial type, module theory, and category theory. We confirm that we have read this submission and believe that we have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however we have significant reservations, as outlined above. reply Respond to this report Responses (0) Higuera Rincon SD and Rubiano Suárez AA. Peer Review Report For: Indeterminacy of Boolean Ring [version 1; peer review: 2 approved with reservations] . F1000Research 2026, 15 :205 ( https://doi.org/10.5256/f1000research.190701.r456955) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. 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