On nil-semicommutative modules

Abstract

In this paper, we introduce the concept of 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 and show that the converse may not be true in general. We also find the conditions under which the quotient module M/N is nil-semicommutative if and only if M is nil-semicommutative. Further, we also prove that for a left R-module M, _RM is nil-semicommutative iff its localization S^{-1}M over the ring S^{-1}R is also nil-semicommutative. Lastly, we explore whether the nil-semicommutative property is preserved under direct product of modules.