Insights into dual Rickart modules: Unveiling the role of second cosingular submodules

Abstract

In this paper, we propose a new type of module by focusing on the second cosingular submodule of a module. We define a module M as weak T-dual Rickart if, for any homomorphism φ ∈ EndR(M), the submodule φ(z̄²(M)) lies above a direct summand of M. We prove that this property is inherited by direct summands of M. We also introduce weak T-dual Baer modules and provide a complete characterization of such modules where the second cosingular submodule is a direct summand. Furthermore, we present a characterization of (semi)perfect rings in which every (finitely generated) module is weak T-dual Rickart.