Subintegrality and ideal class groups of monoid algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918097199300608 |
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| author | Raihan, Md Abu Roberts, Leslie G. Sarwar, Husney Parvez |
| author_facet | Raihan, Md Abu Roberts, Leslie G. Sarwar, Husney Parvez |
| contents | $(1)$ Let $M\subset N$ be a commutative cancellative torsion-free and subintegral extension of monoids. Then we prove that in the case of ring extension $A[M]\subset A[N]$, the two notions, subintegral and weakly subintegral coincide provided $\mathbb{Z}\subset A$. $(2)$ Let $A \subset B$ be an extension of commutative rings and $M\subset N$ an extension of commutative cancellative torsion-free positive monoids. Let $I$ be a radical ideal in $N$. Then $\frac{A[M]}{(I\cap M)A[M]}$ is subintegrally closed in $\frac{B[N]}{IB[N]}$ if and only if the group of invertible $A$-submodules of $B$ is isomorphic to the group of invertible $\frac{A[M]}{(I\cap M)A[M]}$-submodules of $\frac{B[N]}{IB[N]}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_13845 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subintegrality and ideal class groups of monoid algebras Raihan, Md Abu Roberts, Leslie G. Sarwar, Husney Parvez Commutative Algebra 13F15, 13F50, 13B02, 19D45 $(1)$ Let $M\subset N$ be a commutative cancellative torsion-free and subintegral extension of monoids. Then we prove that in the case of ring extension $A[M]\subset A[N]$, the two notions, subintegral and weakly subintegral coincide provided $\mathbb{Z}\subset A$. $(2)$ Let $A \subset B$ be an extension of commutative rings and $M\subset N$ an extension of commutative cancellative torsion-free positive monoids. Let $I$ be a radical ideal in $N$. Then $\frac{A[M]}{(I\cap M)A[M]}$ is subintegrally closed in $\frac{B[N]}{IB[N]}$ if and only if the group of invertible $A$-submodules of $B$ is isomorphic to the group of invertible $\frac{A[M]}{(I\cap M)A[M]}$-submodules of $\frac{B[N]}{IB[N]}$. |
| title | Subintegrality and ideal class groups of monoid algebras |
| topic | Commutative Algebra 13F15, 13F50, 13B02, 19D45 |
| url | https://arxiv.org/abs/2507.13845 |