On the Regularity of Graded k-Algebras of Krull Dimension ...1.
Let be a standard graded -algebra over a field . Then can be written as , where is a graded ideal of a polynomial ring . Assume that and is a strongly stable monomial ideal. We study the symmetric algebra of the first syzygy module of . When the minimal generators of are all of degree 2, the dimension of is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.
In this paper, we deal with the study of quasi-homeomorphisms, the Goldman prime spectrum and the Jacobson prime spectrum of a commutative ring. We prove that, if is a quasi-homeomorphism, a sober space and a continuous map, then there exists a unique continuous map such that . Let be a -space, the injection of onto its sobrification . It is shown, here, that , where is the set of all locally closed points of . Some applications are also indicated. The Jacobson prime spectrum...
Let be a Noetherian ring, and and be two ideals of . Let be a Serre subcategory of the category of -modules satisfying the condition and be a -module. As a generalization of the - and , the - of on is defined as --, and some properties of this concept are investigated. The relations between - and are studied, and it is proved that -, where is a Serre subcategory closed under taking injective hulls. Some conditions are provided that local cohomology modules with...
Let and be commutative rings with identity, be an ideal of , be a ring homomorphism, be an -module, be an -module, and let be an -homomorphism. The amalgamation of with along with respect to denoted by was introduced by M. D’Anna et al. (2010). Recently, R. El Khalfaoui et al. (2021) introduced a special kind of -module called the amalgamation of and along with respect to , and denoted by . We study some homological properties of the -module . Among other results,...
We introduce and study a new class of ring extensions based on a new formula involving the heights of their primes. We compare them with the classical altitude inequality and altitude formula, and we give another characterization of locally Jaffard domains, and domains satisfying absolutely the altitude inequality (resp., the altitude formula). Then we study the extensions R ⊆ S where R satisfies the corresponding condition with respect to S (Definition 3.1). This leads to a new characterization...
Let be a Noetherian local ring and a finitely generated -module. We say has maximal depth if there is an associated prime of such that depth . In this paper we study squarefree monomial ideals which have maximal depth. Edge ideals of cycle graphs, transversal polymatroidal ideals and high powers of connected bipartite graphs with this property are classified.