Some New Classes of Rings Which Have the McCoy Condition
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908337665212416 |
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| author | Danchev, Peter Zahiri, M. |
| author_facet | Danchev, Peter Zahiri, M. |
| contents | We define here the notion of a {\it weakly reversible ring} $R$ saying that a non-zero element $a\in R$ is weakly reversible if there exists an integer $m>0$ depending on $a$ such that $a^m\neq 0$ is reversible, that is, $r_R(a^m)=l_R(a^m)$. In addition, $R$ is weakly reversible if all its elements are weakly reversible. It is shown that all weakly reversible rings are abelian McCoy rings and so, particularly, they are abelian 2-primal rings. Moreover, we construct a weakly reversible ring which is {\it not} reversible. We also show that if $R$ is a weakly reversible ring, then the polynomial ring $R[x]$ is strongly AB. Thus, in particular, the weakly reversible ring $R$ is zip if, and only if, $R[x]$ is zip. We, moreover, prove that if $R$ is a weakly reversible ring and every prime ideal of $R$ is maximal, then both $R$ and $R[x]$ are AB rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18224 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some New Classes of Rings Which Have the McCoy Condition Danchev, Peter Zahiri, M. Rings and Algebras Commutative Algebra 16D15, 16D40, 16D70 We define here the notion of a {\it weakly reversible ring} $R$ saying that a non-zero element $a\in R$ is weakly reversible if there exists an integer $m>0$ depending on $a$ such that $a^m\neq 0$ is reversible, that is, $r_R(a^m)=l_R(a^m)$. In addition, $R$ is weakly reversible if all its elements are weakly reversible. It is shown that all weakly reversible rings are abelian McCoy rings and so, particularly, they are abelian 2-primal rings. Moreover, we construct a weakly reversible ring which is {\it not} reversible. We also show that if $R$ is a weakly reversible ring, then the polynomial ring $R[x]$ is strongly AB. Thus, in particular, the weakly reversible ring $R$ is zip if, and only if, $R[x]$ is zip. We, moreover, prove that if $R$ is a weakly reversible ring and every prime ideal of $R$ is maximal, then both $R$ and $R[x]$ are AB rings. |
| title | Some New Classes of Rings Which Have the McCoy Condition |
| topic | Rings and Algebras Commutative Algebra 16D15, 16D40, 16D70 |
| url | https://arxiv.org/abs/2504.18224 |