拟-中心半交换环

环$R$称为拟-中心半交换的(简称QCS环)如果对$a,b\in R$, $ab=0$蕴含$aRb\subseteq Q(R)$, 其中$Q(R)$为$R$的拟中心.证明了如果$R$ 为QCS环, 那么$R$的幂零元集恰好是它的Wedderburn根, 且对$n\geq 2$, 上三角矩阵环$R=T_n(S)$ 是QCS 环当且仅当$n=2$ 且$S$ 是duo 环, 而$T_{2k+2}^k$是QCS环如果$R$是约化的duo环.

Quasi-Central Semicommutative Rings

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.