On Nil-semicommutative Modules

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.