Commutative rings with $n$-$1$-absorbing prime factorization
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917106429198336 |
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| author | Khalfi, Abdelhaq El Laarabi, Hicham Koç, Suat |
| author_facet | Khalfi, Abdelhaq El Laarabi, Hicham Koç, Suat |
| contents | Let $R$ be a commutative ring with $1\neq 0$ and $n$ be a fixed positive integer. A proper ideal $I$ of $R$ is said to be an \textit{$n$-OA ideal} if whenever $a_1a_2\cdots a_{n+1}\in I$ for some nonunits $a_1,a_2,\ldots,a_{n+1}\in R$, then $a_1a_2\cdots a_n\in I$ or $a_{n+1}\in I$. A commutative ring $R$ is said to be an \textit{$n$-OAF ring} if every proper ideal $I$ of $R$ is a product of finitely many $n$-OA ideals. In fact, $1$-OAF rings and $2$-OAF $2$-OAF-rings are exactly the general ZPI rings and OAF rings, respectively. In addition to giving various properties of $n$-OAF rings, we give a characterization of Noetherian von Neumann regular rings in terms of our new concept. Furthermore, we investigate the $n$-OAF property of some extension of rings such as the polynomial ring $R[X]$, the formal power series ring $R[[X]]$, the ring of $A+XB[X]$, and the trivial extension $R=A\propto E$ of an $A$-module $E$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_21200 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Commutative rings with $n$-$1$-absorbing prime factorization Khalfi, Abdelhaq El Laarabi, Hicham Koç, Suat Commutative Algebra 13B99, 13A15, 13G05, 13F05 Let $R$ be a commutative ring with $1\neq 0$ and $n$ be a fixed positive integer. A proper ideal $I$ of $R$ is said to be an \textit{$n$-OA ideal} if whenever $a_1a_2\cdots a_{n+1}\in I$ for some nonunits $a_1,a_2,\ldots,a_{n+1}\in R$, then $a_1a_2\cdots a_n\in I$ or $a_{n+1}\in I$. A commutative ring $R$ is said to be an \textit{$n$-OAF ring} if every proper ideal $I$ of $R$ is a product of finitely many $n$-OA ideals. In fact, $1$-OAF rings and $2$-OAF $2$-OAF-rings are exactly the general ZPI rings and OAF rings, respectively. In addition to giving various properties of $n$-OAF rings, we give a characterization of Noetherian von Neumann regular rings in terms of our new concept. Furthermore, we investigate the $n$-OAF property of some extension of rings such as the polynomial ring $R[X]$, the formal power series ring $R[[X]]$, the ring of $A+XB[X]$, and the trivial extension $R=A\propto E$ of an $A$-module $E$. |
| title | Commutative rings with $n$-$1$-absorbing prime factorization |
| topic | Commutative Algebra 13B99, 13A15, 13G05, 13F05 |
| url | https://arxiv.org/abs/2511.21200 |