On Nil-semicommutative Modules

TL;DR

This paper introduces the concept of Nil-semicommutative modules, extending the idea from rings to modules, and explores their properties, relationships with other module classes, and conditions for extension and localization.

Contribution

It defines Nil-semicommutative modules, compares them with Weakly semicommutative modules, and investigates their behavior under localization and extension from rings.

Findings

Nil-semicommutative modules are contained in Weakly semicommutative modules

Nil-semicommutative modules do not imply Semicommutative modules and vice versa

Localization preserves Nil-semicommutativity under certain conditions

Abstract

In this paper, we introduce a new concept in Nil-semicommutative modules and present it as an extension of Nil-semicommutative rings to modules. We prove that the class of Nil-semicommutative modules is contained in the class of Weakly semicommutative modules while that of the converse may not be true. We also show that in case of Semicommutative modules and Nil-semicommutative modules, one does not imply the other. Moreover, for a given Nil-semicommutative ring, we provide the conditions under which the same can be extended to a Nil-semicommutative module. Lastly, we also prove that for a left $R$-module $M$, $_RM$ is Nil-semicommutative iff it's localization $S^{-1}M$ over the ring $S^{-1}R$ is also Nil-semicommutative.Various other examples and propositions highlighting the comparative studies of this new class of modules with different classes of modules are also discussed in order to validate the concept.