On strongly semicommutative modules

Abstract

For a left module \( {}_R M \) over a non-commutative ring \( R \), we define the concept of a strongly semicommutative module as a generalization of the reduced module. This notion constitutes a distinct and stronger category within the class of semicommutative modules. We demonstrate that a module \( {}_R M \) is strongly semicommutative if and only if \( {}_{A_n(R)}A_n(M) \) is strongly semicommutative. Additionally, we establish that \( {}_R M \) is strongly semicommutative if and only if \( {}_{R[x]}M[x] \) is strongly semicommutative; this is also equivalent to \( {}_{R[x,x^{-1}]}M[x,x^{-1}] \) being strongly semicommutative. Among our findings, we prove that if \( {}_R M \) is strongly semicommutative, then for any reduced submodule \( N \) of \( M \), the quotient module \( M/N \) is also strongly semicommutative. We provide examples of semicommutative modules that are not strongly semicommutative and show that the class of strongly semicommutative modules remains closed under localization.

Accepted: 11 February 2025