Symmetry over centers. (Q2788831)

As is well-known, a ring is symmetric if for all \(a,b\) and \(c\), \(abc=0\) implies \(acb=0\). Here the authors calls a ring \(R\) (associative with identity) \textit{symmetric-over-center} if \(abc\in C(R)\) implies \(acb\in C(R)\) where \(C(R)\) denotes the center of \(R\). Examples are given to show that this notion is independent to that of a symmetric ring. The canonical example of a symmetric-over-center ring is NEWLINE\[NEWLINED_3(A)=\left\{\left[\begin{smallmatrix} a&b&c\\ 0&a&d\\ 0&0&a\end{smallmatrix}\right]\mid a,b,c,d\in A\right\}NEWLINE\]NEWLINE where \(A\) is a commutative ring. Division rings need not be symmetric-over-center. Various properties of symmetric-over-center rings are given; in particular also for polynomial rings over such rings.