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Volume 27, Issue 2
On Weakly Semicommutative Rings

Weixing Chen & Shuying Cui

Commun. Math. Res., 27 (2011), pp. 179-192.

Published online: 2021-05

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  • Abstract

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.

  • AMS Subject Headings

16U80, 16S50, 16U20, 16N40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-27-179, author = {Chen , Weixing and Cui , Shuying}, title = {On Weakly Semicommutative Rings}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {27}, number = {2}, pages = {179--192}, abstract = {

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} }
TY - JOUR T1 - On Weakly Semicommutative Rings AU - Chen , Weixing AU - Cui , Shuying JO - Communications in Mathematical Research VL - 2 SP - 179 EP - 192 PY - 2021 DA - 2021/05 SN - 27 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19099.html KW - weakly semicommutative ring, weakly 2-primal ring, NI-ring, Armendariz ring. AB -

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.

Weixing Chen & Shuying Cui. (2021). On Weakly Semicommutative Rings. Communications in Mathematical Research . 27 (2). 179-192. doi:
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