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

M. Khudhair Abbas\(^a\), Y. Talebi\(^{b,∗}\), I. Mohammed Ali\(^c\)

\(^a\)Technical College of Management, Baghdad Middle Technical University Baghdad, Iraq

\(^b\)Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

\(^c\)Department of Mathematics, College of Education on Ibn-Al-Haithm, University of Baghdad, Baghdad, Iraq

Correspondence should be addressed to: muntaha2018@mtu.edu.iq, talebi@umz.ac.ir , innam1976@yahoo.com

Copyright © 2024 M. Khudhair Abbas, Y. Talebi, I. Mohammed Ali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Published: Received: 01 August 2023; Accepted: 31 December 2024; Published Online: 31 December 2024

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.

Keywords:

dual Rickart module, T-dual Rickart module, wT d-Rickart module, t-dual Baer module, weak T-dual Baer module.