A ring $R$ is said to be weakly semicommutative if for any $a, b ∈ R$, $ab = 0$ implies $aRb ⊆ {\rm Nil}(R)$, where Nil($R$) is the set of all nilpotent elements in $R$.
In this note, we clarify the relationship between weakly semicommutative rings and
NI-rings by proving that the notion of a weakly semicommutative ring is a proper
generalization of NI-rings. We say that a ring $R$ is weakly 2-primal if the set of
nilpotent elements in $R$ coincides with its Levitzki radical, and prove that if $R$ is a
weakly 2-primal ring which satisfies $α$-condition for an endomorphism $α$ of $R$ (that
is, $ab = 0 ⇔ aα(b) = 0$ where $a, b ∈ R$) then the skew polynomial ring $R[x; α]$ is a weakly 2-primal ring, and that if $R$ is a ring and $I$ is an ideal of $R$ such that $I$ and $R/I$ are both weakly semicommutative then $R$ is weakly semicommutative.
Those extend the main results of Liang et al. 2007 (Taiwanese J. Math., 11(5)(2007),
1359–1368) considerably. Moreover, several new results about weakly semicommutative rings and NI-rings are included.
