Commutative Self-Injective Rings

1970 ◽  
Vol 22 (6) ◽  
pp. 1101-1108 ◽  
Author(s):  
Surjeet Singh ◽  
Kamlesh Wasan
Keyword(s):  
Prime Ideal ◽  
Homomorphic Image ◽  
Noetherian Rings

All rings considered here are commutative containing at least two elements, but may not have identity. A ring R is said to be selfinjective if R as an R-module is injective. A ring R is said to be pre-selfinjective if every proper homomorphic image of R is self-injective [9]. Study of pre-self-injective rings was initiated by Levy [10], who established a characterization of Noetherian pre-self-injective rings with identity in terms of other well-known types of rings. Recently Klatt and Levy [9] have characterized all pre-self-injective rings with identity. In this paper we are mainly interested in Noetherian rings. For the sake of convenience we shall call a pre-self-injective ring an (I)-ring. A ring R will be said to be a (PMI)-ring if for each proper prime ideal P with P2 ≠ 0, the ring R/P2 is self-injective. Clearly, an (I)-ring is a (PMI)-ring.

scholarly journals A characterization of minimal prime ideals

1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Sheaf Representations ◽  
Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

On copure projective and cotorsion modules

2019 ◽  
Vol 56 (1) ◽  
pp. 1-12
Author(s):  
Wei Ren ◽  
Duocai Zhang
Keyword(s):  
Noetherian Rings ◽  
Derived Functor ◽  
Flat Modules

Abstract Let R be an IF ring, or be a ring such that each right R-module has a monomorphic flat envelope and the class of flat modules is coresolving. We firstly give a characterization of copure projective and cotorsion modules by lifting and extension diagrams, which implies that the classes of copure projective and cotorsion modules have some balanced properties. Then, a relative right derived functor is introduced to investigate copure projective and cotorsion dimensions of modules. As applications, some new characterizations of QF rings, perfect rings and noetherian rings are given.

Rings with A Finitely Generated Total Quotient Ring

1974 ◽  
Vol 17 (1) ◽  
pp. 1-4 ◽  
Author(s):  
John Conway Adams
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Quotient Ring ◽  
Finitely Generated ◽  
Zero Divisors ◽  
Rank One ◽  
Definition Of

Let R be a commutative ring with non-zero identity and let K be the total quotient ring of R. We call R a G-ring if K is finitely generated as a ring over R. This generalizes Kaplansky′s definition of G-domain [5].Let Z(R) be the set of zero divisors in R. Following [7] elements of R—Z(R) and ideals of R containing at least one such element are called regular. Artin-Tate's characterization of Noetherian G-domains [1, Theorem 4] carries over with a slight adjustment to characterize a Noetherian G-ring as being semi-local in which every regular prime ideal has rank one.

scholarly journals A characterization of noetherian rings by cyclic modules

1996 ◽  
Vol 39 (2) ◽  
pp. 253-262 ◽  
Author(s):  
Dinh Van Huynh
Keyword(s):  
Projective Module ◽  
Noetherian Rings ◽  
Noetherian Module ◽  
Direct Sum

It is shown that a ring R is right noetherian if and only if every cyclic right R-module is injective or a direct sum of a projective module and a noetherian module.

scholarly journals On the upper dual Zariski topology

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 483-489
Author(s):  
Seçil Çeken
Keyword(s):  
Prime Ideal ◽  
Compact Space ◽  
Spectral Space ◽  
Zariski Topology ◽  
Algebraic Properties ◽  
Patch Topology ◽  
Totally Disconnected ◽  
Second Spectrum

Let R be a ring with identity and M be a left R-module. The set of all second submodules of M is called the second spectrum of M and denoted by Specs(M). For each prime ideal p of R we define Specsp(M) := {S? Specs(M) : annR(S) = p}. A second submodule Q of M is called an upper second submodule if there exists a prime ideal p of R such that Specs p(M)? 0 and Q = ? S2Specsp(M)S. The set of all upper second submodules of M is called upper second spectrum of M and denoted by u.Specs(M). In this paper, we discuss the relationships between various algebraic properties of M and the topological conditions on u.Specs(M) with the dual Zarsiki topology. Also, we topologize u.Specs(M) with the patch topology and the finer patch topology. We show that for every left R-module M, u.Specs(M) with the finer patch topology is a Hausdorff, totally disconnected space and if M is Artinian then u.Specs(M) is a compact space with the patch and finer patch topology. Finally, by applying Hochster?s characterization of a spectral space, we show that if M is an Artinian left R-module, then u.Specs(M) with the dual Zariski topology is a spectral space.

scholarly journals Smoothness of Noetherian rings

1984 ◽  
Vol 95 ◽  
pp. 163-179 ◽  
Author(s):  
Hiroshi Tanimoto
Keyword(s):  
Noetherian Rings ◽  
List Type ◽  
The Difference

In [16] we studied the following problems which had been asked by H. Matsumura (cf. [11]): (I)What is the difference between smoothness and J-smoothness? In particular, concerning the characterization of smoothness,(II)When is a ring A[X1 …, Xn]/a smooth over A?In this paper, according to these problems, we will study J-smoothness further when rings are noetherian as in [16].

Graded 1-absorbing prime ideals of graded commutative rings

2021 ◽  
Author(s):  
Mohammed Issoual
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Graded Ring

Let [Formula: see text] be a group with identity [Formula: see text] and [Formula: see text] be [Formula: see text]-graded commutative ring with [Formula: see text] In this paper, we introduce and study the graded versions of 1-absorbing prime ideal. We give some properties and characterizations of these ideals in graded ring, and we give a characterization of graded 1-absorbing ideal the idealization [Formula: see text]

scholarly journals Reflexive ideals and injective modules over Noetherian v-H orders

1991 ◽  
Vol 34 (1) ◽  
pp. 31-43 ◽  
Author(s):  
K. A. Brown ◽  
A. Haghany ◽  
T. H. Lenagan
Keyword(s):  
Prime Ideal ◽  
Recent Work ◽  
Torsion Theory ◽  
Noetherian Rings ◽  
Prime Rings ◽  
Injective Modules ◽  
Integrally Closed

The class of prime Noetherian v-H orders is a class of Noetherian prime rings including the commutative integrally closed Noetherian domains, and the hereditary Noetherian prime rings, and designed to mimic the latter at the level of height one primes. We continue recent work on the structure of indecomposable injective modules over Noetherian rings by describing the structure of such a module E over a prime Noetherian v-H order R in the case where the assassinator P of E is a reflexive prime ideal. This description is then applied to a problem in torsion theory, so generalising work of Beck, Chamarie and Fossum.

scholarly journals Characterization of the automorphisms having the lifting property in the category of abelianp-groups

2003 ◽  
Vol 2003 (71) ◽  
pp. 4511-4516
Author(s):  
S. Abdelalim ◽  
H. Essannouni
Keyword(s):  
Homomorphic Image ◽  
Abelian Groups ◽  
Torsion Group ◽  
Divisible Group ◽  
Torsion Free

Letpbe a prime. It is shown that an automorphismαof an abelianp-groupAlifts to any abelianp-group of whichAis a homomorphic image if and only ifα=π idA, withπan invertiblep-adic integer. It is also shown that ifAis torsion group or torsion-freep-divisible group, thenidAand−idAare the only automorphisms ofAwhich possess the lifting property in the category of abelian groups.

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