Module-theoretic Bott-Duffin and outer generalized invertibility

Abstract

In this paper, we introduce and study two generalized inverse concepts in module theory, extending the well-established theory of generalized inverses in rings: the Bott-Duffin $(e,f)$-inverse and the $(e,f)$-outer generalized inverse. By extending established ring-theoretic definitions, we prove the unique existence of both inverses under specific conditions. We derive necessary and sufficient criteria for their existence and investigate their fundamental properties.