{\Sigma}-dual Rickart modules

TL;DR

This paper introduces and characterizes { extSigma}-dual Rickart modules, exploring their properties, connections with various ring types, and the structure of their endomorphism rings, expanding the theory of dual Rickart modules.

Contribution

It defines { extSigma}-dual Rickart modules, characterizes them via strongly cogenerated modules, and investigates their properties and relationships with different classes of rings.

Findings

Every cohereditary module over a Noetherian ring is { extSigma}-dual Rickart.

{ extSigma}-dual Rickart modules relate to semisimple Artinian, regular, and semi-hereditary rings.

Properties of endomorphism rings of { extSigma}-dual Rickart modules are studied.

Abstract

In this paper, we dualize the concept of {\Sigma}-Rickart modules as {\Sigma}-dual Rickart modules. An R-module M is said to be {\Sigma}-dual Rickart if the direct sum of arbitrary copies of M is dual Rickart. We prove that each cohereditary module over the Noetherian ring is a {\Sigma}-dual Rickart. We introduce the notion of strongly cogenerated modules and characterize {\Sigma}-dual Rickart modules in terms of strongly cogenerated modules. We also study some properties of {\Sigma}- dual Rickart modules and find connections with semisimple Artinian ring, regular ring semi-hereditary ring and FP-injective module. Further, we study the endomorphism ring of {\Sigma}-dual Rickart modules