Subintegrality and ideal class groups of monoid algebras

2025/07/18 by Md Abu Raihan, Raihan, Md Abu, Leslie G. Roberts +3
Computer Science · Mathematics · #13B02 #13F15 #13F50 #19D45 #Advanced Algebra and Logic #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2507.13845

openalex publication_date 2025/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

(1) Let M⊂ N be a commutative cancellative torsion-free and subintegral extension of monoids. Then we prove that in the case of ring extension A[M]⊂ A[N], the two notions, subintegral and weakly subintegral coincide provided ℤ⊂ A. (2) Let A ⊂ B be an extension of commutative rings and M⊂ N an extension of commutative cancellative torsion-free positive monoids. Let I be a radical ideal in N. Then (A[M])/((I∩ M)A[M]) is subintegrally closed in (B[N])/(IB[N]) if and only if the group of invertible A-submodules of B is isomorphic to the group of invertible (A[M])/((I∩ M)A[M])-submodules of (B[N])/(IB[N]).

Citations

• Unimodular rows over monoid rings
• The K-theory of toric schemes over regular rings of mixed characteristic
• Ideal class groups of monoid algebras
• Subintegrality, Invertible Modules and Laurent Polynomial Extensions
• The nilpotence conjecture in K-theory of toric varieties

Related

• Ideal class groups of monoid algebras
• Weak subintegral closure of ideals
• IE-closed subcategories of commutative rings are torsion-free classes
• Subset expansions of monoids
• Unimodular rows over monoid extensions of overrings of polynomial rings