BT Inverse and the Generalized BT Inverse in a Ring

In this paper, we introduce and investigate the BT inverse for a ring element. The binary relation BT order has also been studied. Furthermore, we introduce the generalized BT inverse and present some presentations of this new generalized inverse. It also be characterized by using system of equations and Pierce decompositions of ring elements. Many properties of BT inverse for complex matrices are generalized to a broader context within a ring.