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Cohen-Macaulayness of multiplication rings and modules

R. Naghipour, H. Zakeri, N. Zamani (2003)

Colloquium Mathematicae

Let R be a commutative multiplication ring and let N be a non-zero finitely generated multiplication R-module. We characterize certain prime submodules of N. Also, we show that N is Cohen-Macaulay whenever R is Noetherian.

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...

Conditions under which R ( x ) and R 〈 x 〉 are almost Q-rings

Hani A. Khashan, H. Al-Ezeh (2007)

Archivum Mathematicum

All rings considered in this paper are assumed to be commutative with identities. A ring R is a Q -ring if every ideal of R is a finite product of primary ideals. An almost Q -ring is a ring whose localization at every prime ideal is a Q -ring. In this paper, we first prove that the statements, R is an almost Z P I -ring and R [ x ] is an almost Q -ring are equivalent for any ring R . Then we prove that under the condition that every prime ideal of R ( x ) is an extension of a prime ideal of R , the ring R is a (an almost)...

Conducteur, descente et pincement

Daniel Ferrand (2003)

Bulletin de la Société Mathématique de France

Une somme amalgamée de schémas est décrite localement par un produit fibré d’anneaux. Ce texte donne un résultat global d’existence (§5.4) de schémas définis comme certaines sommes amalgamées et un procédé algébrique (§2.2) pour décrire les modules sur produits fibrés d’anneaux correspondants.

Congruences and ideals in ternary rings

Ivan Chajda, Radomír Halaš, František Machala (1997)

Czechoslovak Mathematical Journal

A ternary ring is an algebraic structure ℛ = ( R ; t , 0 , 1 ) of type ( 3 , 0 , 0 ) satisfying the identities t ( 0 , x , y ) = y = t ( x , 0 , y ) and t ( 1 , x , 0 ) = x = ( x , 1 , 0 ) where, moreover, for any a , b , c ∈ R there exists a unique d ∈ R with t ( a , b , d ) = c . A congruence θ on ℛ is called normal if ℛ / θ is a ternary ring again. We describe basic properties of the lattice of all normal congruences on ℛ and establish connections between ideals (introduced earlier by the third author) and congruence kernels.

Contracting endomorphisms and dualizing complexes

Saeed Nasseh, Sean Sather-Wagstaff (2015)

Czechoslovak Mathematical Journal

We investigate how one can detect the dualizing property for a chain complex over a commutative local Noetherian ring R . Our focus is on homological properties of contracting endomorphisms of R , e.g., the Frobenius endomorphism when R contains a field of positive characteristic. For instance, in this case, when R is F -finite and C is a semidualizing R -complex, we prove that the following conditions are equivalent: (i) C is a dualizing R -complex; (ii) C ∼ 𝐑 Hom R ( n R , C ) for some n > 0 ; (iii) G C -dim n R < ∞ and C is derived 𝐑 Hom R ( n R , C ) -reflexive...

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