A Unified Approach to Generalizing π-Extending and π-Baer Rings

This paper introduces and examines the right essentially π-Baer ring property, which serves as a new extension of the π-extending and π-Baer ring conditions. The initial phase of the study involves the development of several foundational results. The subsequent phase of the study involves the exploration of the transfer of the right essentially π-Baer condition between a ring R and its various ring extensions. The paper provides a characterization of the right essentially π-Baer generalized triangular matrix rings, as well as n×n upper triangular matrix rings of R and a specific type of trivial extension of R. The findings are illustrated by means of several examples.