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Combinatoric of syzygies for semigroup algebras.

Emilio Briales, Pilar Pisón, Antonio Campillo, Carlos Marijuán (1998)

Collectanea Mathematica

We describe how the graded minimal resolution of certain semigroup algebras is related to the combinatorics of some simplicial complexes. We obtain characterizations of the Cohen-Macaulay and Gorenstein conditions. The Cohen-Macaulay type is computed from combinatorics. As an application, we compute explicitly the graded minimal resolution of monomial both affine and simplicial projective surfaces.

Cominimaxness of local cohomology modules

Moharram Aghapournahr (2019)

Czechoslovak Mathematical Journal

Let R be a commutative Noetherian ring, I an ideal of R . Let t ∈ ℕ 0 be an integer and M an R -module such that Ext R i ( R / I , M ) is minimax for all i ≤ t + 1 . We prove that if H I i ( M ) is FD ≤ 1 (or weakly Laskerian) for all i < t , then the R -modules H I i ( M ) are I -cominimax for all i < t and Ext R i ( R / I , H I t ( M ) ) is minimax for i = 0 , 1 . Let N be a finitely generated R -module. We prove that Ext R j ( N , H I i ( M ) ) and Tor j R ( N , H I i ( M ) ) are I -cominimax for all i and j whenever M is minimax and H I i ( M ) is FD ≤ 1 (or weakly Laskerian) for all i .

Commutative graded- S -coherent rings

Anass Assarrar, Najib Mahdou, Ünsal Tekir, Suat Koç (2023)

Czechoslovak Mathematical Journal

Recently, motivated by Anderson, Dumitrescu’s S -finiteness, D. Bennis, M. El Hajoui (2018) introduced the notion of S -coherent rings, which is the S -version of coherent rings. Let R = ⨁ α ∈ G R α be a commutative ring with unity graded by an arbitrary commutative monoid G , and S a multiplicatively closed subset of nonzero homogeneous elements of R . We define R to be graded- S -coherent ring if every finitely generated homogeneous ideal of R is S -finitely presented. The purpose of this paper is to give the graded...

Commutative rings whose certain modules decompose into direct sums of cyclic submodules

Farid Kourki, Rachid Tribak (2023)

Czechoslovak Mathematical Journal

We provide some characterizations of rings R for which every (finitely generated) module belonging to a class 𝒞 of R -modules is a direct sum of cyclic submodules. We focus on the cases, where the class 𝒞 is one of the following classes of modules: semiartinian modules, semi-V-modules, V-modules, coperfect modules and locally supplemented modules.

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