Quasi-Central Semicommutative Rings

Abstract:

      A ring $R$ is said to be quasi-central semicommutative (simply, a QCS ring) if $ab=0$ implies $aRb\subseteq Q(R)$ for $a,b\in R$, where $Q(R)$ is the quasi-center of $R$. It is proved that if $R$ is a QCS ring, then the set of nilpotent elements of $R$ coincides with its Wedderburn radical, and that the upper triangular matrix ring $R=T_n(S)$ for any $n\geq 2$ is a QCS ring if and only if $n=2$ and $S$ is a duo ring, while $T_{2k+2}^k(R)$ is a QCS ring when $R$ is a reduced duo ring.