On Strongly Symmetric Rings

Abstract

A ring $R$ is called strongly symmetric, if whenever polynomials $f(x), g(x), h(x)$ in $R[x]$ satisfy $f(x)g(x)h(x) = 0,$ then $f(x)h(x)g(x) = 0.$ It is proved that a ring $R$ is strongly symmetric if and only if its polynomial ring $R[x]$ is strongly symmetric if and only if its Laurent polynomial ring $R[x, x^{-1}]$ is strongly symmetric. We also show that for a right Ore ring $R$ with $Q$ its classical right quotient ring, $R$ is strongly symmetric if and only if $Q$ is strongly symmetric. Finally we proved that, let $R$ be an algebra over a commutative ring $S,$ and $D$ be the Dorroh extension of $R$ by $S.$ If $R$ is strongly symmetric and $S$ is a domain, then $D$ is strongly symmetric.