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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.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19099.html} }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.

*Communications in Mathematical Research*.

*27*(2). 179-192. doi: