Product of prime ideals as factorization of submodules
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912693768683520 |
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| author | Thulasi, K. R. Duraivel, T. Mangayarcarassy, S. |
| author_facet | Thulasi, K. R. Duraivel, T. Mangayarcarassy, S. |
| contents | For a proper submodule $N$ of a finitely generated module $M$ over a Noetherian ring, the product of prime ideals which occur in a regular prime extension filtration of $M$ over $N$ is defined as its generalized prime ideal factorization in $M$. In this article, we find conditions for a product of prime ideals to be the generalized prime ideal factorization of a submodule of some module. We show that a power of a prime ideal occurs in a generalized prime ideal factorization only if it is not equal to its lesser powers. Also, we show that ${\mathfrak{p}_1}^{r_1} \cdots {\mathfrak{p}_{n}}^{r_{n}}$ is a generalized prime ideal factorization if and only if for each $1 \leq i \leq n$, ${\mathfrak{p}_i}^{r_i}$ is the generalized prime ideal factorization of some submodule of a module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_14285 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Product of prime ideals as factorization of submodules Thulasi, K. R. Duraivel, T. Mangayarcarassy, S. Commutative Algebra Primary: 13A05, Secondary: 13A15, 13E05, 13E15 For a proper submodule $N$ of a finitely generated module $M$ over a Noetherian ring, the product of prime ideals which occur in a regular prime extension filtration of $M$ over $N$ is defined as its generalized prime ideal factorization in $M$. In this article, we find conditions for a product of prime ideals to be the generalized prime ideal factorization of a submodule of some module. We show that a power of a prime ideal occurs in a generalized prime ideal factorization only if it is not equal to its lesser powers. Also, we show that ${\mathfrak{p}_1}^{r_1} \cdots {\mathfrak{p}_{n}}^{r_{n}}$ is a generalized prime ideal factorization if and only if for each $1 \leq i \leq n$, ${\mathfrak{p}_i}^{r_i}$ is the generalized prime ideal factorization of some submodule of a module. |
| title | Product of prime ideals as factorization of submodules |
| topic | Commutative Algebra Primary: 13A05, Secondary: 13A15, 13E05, 13E15 |
| url | https://arxiv.org/abs/2308.14285 |