On strongly semicommutative modules

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.