2025/12/24 by Tony J. Puthenpurakal, Puthenpurakal, Tony J., Samarendra Sahoo +1
Mathematics · #Commutative Algebra and Its Applications #Algebraic structures and combinatorial models #Rings, Modules, and Algebras
paper · doi:10.48550/arxiv.2512.21168
Let A be a non Gorenstein Cohen Macaulay ring of dimension d≥ 1, I an ideal of A, and suppose ωA is a canonical A-module. Set r(I,ωA) = \bigcupn ≥ 0 (In+1 ωA : In ωA) ⊆ A . We show that the ideal r(I,-) is ωA invariant. Motivated by this property, we introduce a new class of rings, which we call quasi Hilbert rings. We provide several examples of quasi Hilbert rings and discuss a number of their applications. Let A be a local ring with maximal ideal \mathfrakm. We prove that A is quasi Hilbert iff \widehatA is quasi Hilbert, where \widehatA is the completion of A w.r.t. \mathfrakm. If d≥ 2 and x∈ \mathfrakm∖ \mathfrakm2 is an A\bigoplus ωA superficial element, we prove that if A is quasi Hilbert, then so is A/(x). Writing \widetildeI for the Ratliff Rush closure of an ideal I, we also provide sufficient conditions ensuring the vanishing of r(In,ωA)/\widetildeIn for all n≥ 1.