Pointwise Semicommutative Rings

We call a ring $R$ pointwise semicommutative if for any element $a\in R$ either $l(a)$ or $r(a)$ is an ideal of $R$. The class of pointwise semicommutative rings is a strict generalization of semicommutative rings. Since reduced rings are pointwise semicommutative, this paper studies sufficient conditions for pointwise semicommutative rings to be reduced. For a pointwise semicommutative ring $R$, $R$ is strongly regular if and only if $R$ is left SF; $R$ is exchange if and only if $R$ is clean; if $R$ is semiperiodic then $R/J(R)$ is commutative.