Regularity of Ideals and their Radicals.
We study the construction of new multiplication modules relative to a torsion theory . As a consequence, -finitely generated modules over a Dedekind domain are completely determined. We relate the relative multiplication modules to the distributive ones.
Let be a commutative ring with identity. A proper ideal is said to be an -ideal of if for , and imply . We give a new generalization of the concept of -ideals by defining a proper ideal of to be a semi -ideal if whenever is such that , then or . We give some examples of semi -ideal and investigate semi -ideals under various contexts of constructions such as direct products, homomorphic images and localizations. We present various characterizations of this new class of...
The notion of a d-sequence in Commutative Algebra was introduced by Craig Huneke, while the notion of a sequence of linear type was introduced by Douglas Costa. Both types of sequences generate ideals of linear type. In this paper we study another type of sequences, that we call c-sequences. They also generate ideals of linear type. We show that c-sequences are in between d-sequences and sequences of linear type and that the initial subsequences of c-sequences are c-sequences. Finally we prove a...
Soient un corps commutatif et un idéal de l’anneau des polynômes (éventuellement ). Nous prouvons une conjecture de C. Berenstein - A. Yger qui affirme que pour tout polynôme , élément de la clôture intégrale de l’idéal , on a une représentationoù .
Let be a commutative ring with identity and be the set of ideals with nonzero annihilator. The strongly annihilating-ideal graph of is defined as the graph with the vertex set and two distinct vertices and are adjacent if and only if and . In this paper, the perfectness of for some classes of rings is investigated.
The Dedekind-Mertens lemma relates the contents of two polynomials and the content of their product. Recently, Epstein and Shapiro extended this lemma to the case of power series. We review the problem with a special emphasis on the case of power series, give an answer to a question posed by Epstein-Shapiro and investigate extensions of some related results. This note is of expository character and discusses the history of the problem, some examples and announces some new results.