An equivalent condition and some properties of strong \(J\)-symmetric ring (Q2052110)

Summary: Let \(J(R)\) denote the Jacobson radical of a ring \(R\). We say that ring \(R\) is strong \(J\)-symmetric if, for any \(a,b,c\in R\), \(abc\in J(R)\) implies \(bac\in J(R)\). If ring \(R\) is strong \(J\)-symmetric, then it is proved that \(R[x]/(x^n)\) is strong \(J\)-symmetric for any \(n\geq2\). If \(R\) and \(S\) are rings and \(_R W_S\) is a \((R, S)\)-bimodule, \(E=T(R, S, W)=\begin{pmatrix} R & W \\ 0 & S \end{pmatrix}=\left\{\begin{pmatrix} r & w \\ 0 & s \end{pmatrix}\mid r \in R, w \in W, s \in S\right\}\), then it is proved that \(R\) and \(S\) are \(J\)-symmetric if and only if \(E\) is \(J\)-symmetric. It is also proved that \(R\) and \(S\) are strong \(J\)-symmetric if and only if \(E\) is strong \(J\)-symmetric.