On strongly semicommutative modules

Source 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