Injective modules and classical localization in Noetherian rings

Transactions of the American Mathematical Society, 1974

Contemporary Mathematics, 2018

We determine the universal localization R Γ(S) at certain semiprime ideals S of a Noetherian ring R that is a finite extension of its center. For this class of semiprime ideals, R Γ(S) is Noetherian, which need not be true in the general case.

Canadian Journal of Mathematics, 1980

The main object of this paper is to study when infective noetherian modules are artinian. This question was first raised by J. Fisher and an example of an injective noetherian module which is not artinian is given in [9]. However, it is shown in [4] that over commutative rings, and over hereditary noetherian P.I rings, injective noetherian does imply artinian. By combining results of [6] and [4] it can be shown that the above implication is true over any noetherian P.I ring. It is shown in this paper that injective noetherian modules are artinian over rings finitely generated as modules over their centers, and over semiprime rings of Krull dimension 1. It is also shown that every injective noetherian module over a P.I ring contains a simple submodule. Since any noetherian injective module is a finite direct sum of indecomposable injectives it suffices to study when such injectives are artinian. If Q is an injective indecomposable noetherian module, then Q contains a non-zero submodu...

Journal of Algebra and Its Applications, 2015

In this paper we present a technical lemma about localization at countably infinitely many prime ideals. We apply this lemma to get many results about the finiteness of associated prime ideals of local cohomology modules.

International Journal of Algebra, 2015

In this paper some results that concerning localization of commutative rings and modules are proved. It also, studies the effect of localization on certain types of ideals and modules such as G−ideals, G−submodules, G−weakly submodules and G−modules. Several conditions are given under which certain properties of such types of algebraic structures are preserved under localization.

Glasgow Mathematical Journal, 1995

Throughout this paper R will be an associative ring with unity and all R-modules are unitary. The right (resp. left) annihilator in R of a subset X of a module is denoted by r(X)(resp. I(X)). The Jacobson radical of R is denoted by J(R), the singular ideals are denoted by Z(RR) and Z(RR) and the socles by Soc(RR) and Soc(RR). For a module M, E(M) and PE(M) denote the injective and pure-injective envelopes of M, respectively. For a submodule A ⊆ M, the notation A ⊆⊕M will mean that A is a direct summand of M.

Communications in Algebra, 2003

Let R be a commutative ring with 1 such that Nil(R) is a divided prime ideal of R. The purpose of this paper is to introduce a new class of rings that is closely related to the class of Noetherian rings. A ring R is called a Nonnil-Noetherian ring if every nonnil ideal of R is finitely generated. We show that many of the properties of Noetherian rings are also true for Nonnil-Noetherian rings; we use the idealization construction to give examples of Nonnil-Noetherian rings that are not Noetherian rings; we show that for each n ! 1, there is a Nonnil-Noetherian ring with Krull dimension n which is not a Noetherian ring.

Forum Mathematicum

We develop a technique to construct finitely injective modules which are non trivial, in the sense that they are not direct sums of injective modules. As a consequence, we prove that a ring $R$ is left noetherian if and only if each finitely injective left $R$-module is trivial, thus answering an open question posed by Salce.

2003

Let R be a commutative ring with 1 such that Nil(R) is a divided prime ideal of R. The purpose of this paper is to introduce a new class of rings that is closely related to the class of Noetherian rings. A ring R is called a Nonnil-Noetherian ring if every nonnil ideal of R is finitely generated. We show that many of the properties of Noetherian rings are also true for Nonnil-Noetherian rings; we use the idealization construction to give examples of Nonnil-Noetherian rings that are not Noetherian rings; we show that for each n ! 1, there is a Nonnil-Noetherian ring with Krull dimension n which is not a Noetherian ring.

Transactions of the American Mathematical Society, 1971

This paper centers around the theorem that a commutative ring R is noetherian if every RP, P prime, is noetherian and every finitely generated ideal of R has only finitely many weak-Bourbaki associated primes. A more precise local version of this theorem is also given, and examples are presented to show that the assumption on the weak-Bourbaki primes cannot be deleted nor replaced by the assumption that Spec (Ä) is noetherian. Moreover, an alternative statement of the theorem using a variation of the weak-Bourbaki associated primes is investigated. The proof of the theorem involves a knowledge of the behavior of associated primes of an ideal under quotient ring extension, and the paper concludes with some remarks on this behavior in the more general setting of flat ring extensions.