Rigid, quasi-rigid and matrix rings with $(\overline{\sigma},0)$-multiplication

Let $R$ be a ring with an endomorphism $\sigma$. We introduce $(\overline{\sigma}, 0)$-multiplication which is a generalization of the simple $ 0$-multiplication. It is proved that for arbitrary positive integers $m\leq n$ and $n\geq 2$, $R[x; \sigma]$ is a reduced ring if and only if $S_{n, m}(R)$ is a ring with $(\overline{\sigma},0)$-multiplication.