scholarly journals Secondary representations for injective modules over commutative Noetherian rings

1976 ◽  
Vol 20 (2) ◽  
pp. 143-151 ◽  
Author(s):  
Rodney Y. Sharp
Keyword(s):  
Representation Theory ◽  
Commutative Ring ◽  
Noetherian Rings ◽  
Primary Decomposition ◽  
Dual Theory ◽  
Injective Modules ◽  
Finite Sum ◽  
Secondary Representation

There have been several recent accounts of a theory dual to the well-known theory of primary decomposition for modules over a (non-trivial) commutative ring A with identity: see (4), (2) and (9). Here we shall follow Macdonald's terminology from (4) and refer to this dual theory as “ secondary representation theory ”. A secondary representation for an A-module M is an expression for M as a finite sum of secondary submodules; just as the zero submodule of a Noetherian A-module X has a primary decomposition in X, it turns out, as one would expect, that every Artinian A-module has a secondary representation.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings

1991 ◽  
Vol 34 (1) ◽  
pp. 155-160 ◽  
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

LetEbe an injective module over the commutative Noetherian ringA, and letabe an ideal ofA. TheA-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈Nis ultimately constant. This result is analogous to a theorem of M. Brodmann that, ifMis a finitely generatedA-module, then the sequence of sets (AssA(M/αnM))n∈Nis ultimately constant.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings II

1992 ◽  
Vol 35 (3) ◽  
pp. 511-518
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Integral Closure ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

Let E be an injective module over a commutative Noetherian ring A (with non-zero identity), and let a be an ideal of A. The submodule (0:Eα) of E has a secondary representation, and so we can form the finite set AttA(0:Eα) of its attached prime ideals. In [1, 3.1], we showed that the sequence of sets is ultimately constant; in [2], we introduced the integral closure a*(E) of α relative to E, and showed that is increasing and ultimately constant. In this paper, we prove that, if a contains an element r such that rE = E, then is ultimately constant, and we obtain information about its ultimate constant value.

Envelopes, covers and semidualizing modules

2019 ◽  
Vol 18 (07) ◽  
pp. 1950137
Author(s):  
Lixin Mao
Keyword(s):  
Commutative Ring ◽  
Noetherian Rings ◽  
Artinian Rings ◽  
Injective Modules ◽  
Semidualizing Modules ◽  
Coherent Rings ◽  
The Relationship

Given an [Formula: see text]-module [Formula: see text] and a class of [Formula: see text]-modules [Formula: see text] over a commutative ring [Formula: see text], we investigate the relationship between the existence of [Formula: see text]-envelopes (respectively, [Formula: see text]-covers) and the existence of [Formula: see text]-envelopes or [Formula: see text]-envelopes (respectively, [Formula: see text]-covers or [Formula: see text]-covers) of modules. As a consequence, we characterize coherent rings, Noetherian rings, perfect rings and Artinian rings in terms of envelopes and covers by [Formula: see text]-projective, [Formula: see text]-flat, [Formula: see text]-injective and [Formula: see text]-[Formula: see text]-injective modules, where [Formula: see text] is a semidualizing [Formula: see text]-module.

Strict Mittag–Leffler modules over Gorenstein injective modules

2019 ◽  
Vol 19 (03) ◽  
pp. 2050050 ◽  
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan
Keyword(s):  
Noetherian Ring ◽  
Direct Limit ◽  
Injective Module ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Injective Modules ◽  
Pure Submodules ◽  
Covering Class ◽  
Gorenstein Injective ◽  
Finitely Presented

In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class [Formula: see text] of Gorenstein injective modules. To this aim, we introduce the notion of [Formula: see text]-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented [Formula: see text]-projective modules, then it is a strict Mittag–Leffler module over [Formula: see text]. As applications, we prove that if [Formula: see text] is a two-sided noetherian ring, then [Formula: see text] is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over [Formula: see text].

Definability problems for modules and rings

1971 ◽  
Vol 36 (4) ◽  
pp. 623-649 ◽  
Author(s):  
Gabriel Sabbagh ◽  
Paul Eklof
Keyword(s):  
Ring Theory ◽  
Noetherian Rings ◽  
Fixed Ring ◽  
Injective Modules ◽  
Free Left ◽  
Right Noetherian Rings ◽  
The Right

This paper is concerned with questions of the following kind: let L be a language of the form Lαω and let be a class of modules over a fixed ring or a class of rings; is it possible to define in L? We will be mainly interested in the cases where L is Lωω or L∞ω and is a familiar class in homologic algebra or ring theory.In Part I we characterize the rings Λ such that the class of free (respectively projective, respectively flat) left Λ-modules is elementary. In [12] we solved the corresponding problems for injective modules; here we show that the class of injective Λ-modules is definable in L∞ω if and only if it is elementary. Moreover we identify the right noetherian rings Λ such that the class of projective (respectively free) left Λ-modules is definable in L∞ω.

SUPERDECOMPOSABLE PURE INJECTIVE MODULES OVER COMMUTATIVE NOETHERIAN RINGS

2008 ◽  
Vol 07 (06) ◽  
pp. 809-830
Author(s):  
VERA PUNINSKAYA ◽  
CARLO TOFFALORI
Keyword(s):  
Noetherian Rings ◽  
Injective Modules ◽  
Gabriel Dimension

We investigate width and Krull–Gabriel dimension over commutative Noetherian rings which are "tame" according to the Klingler–Levy analysis in [4–6], in particular over Dedekind-like rings and their homomorphic images. We show that both are undefined in most cases.

INJECTIVE CLASSES OF MODULES

2013 ◽  
Vol 12 (04) ◽  
pp. 1250188 ◽  
Author(s):  
WOJCIECH CHACHÓLSKI ◽  
WOLFGANG PITSCH ◽  
JÉRÔME SCHERER
Keyword(s):  
Commutative Ring ◽  
Homological Algebra ◽  
Injective Modules

We study classes of modules over a commutative ring which allow to do homological algebra relative to such a class. We classify those classes consisting of injective modules by certain subsets of ideals. When the ring is Noetherian the subsets are precisely the generization closed subsets of the spectrum of the ring.

REPRESENTATION THEORY FOR VARIETIES OF COMTRANS ALGEBRAS AND LIE TRIPLE SYSTEMS

2011 ◽  
Vol 21 (03) ◽  
pp. 459-472 ◽  
Author(s):  
BOKHEE IM ◽  
JONATHAN D. H. SMITH
Keyword(s):  
Representation Theory ◽  
Commutative Ring ◽  
Triple Systems ◽  
Enveloping Algebra ◽  
Representations Of Algebras

For a variety of comtrans algebras over a commutative ring, representations of algebras in the variety are identified as modules over an enveloping algebra. In particular, a new, simpler approach to representations of Lie triple systems is provided.

C3-Modules

2015 ◽  
Vol 22 (04) ◽  
pp. 655-670 ◽  
Author(s):  
Ismail Amin ◽  
Yasser Ibrahim ◽  
Mohamed Yousif
Keyword(s):  
Commutative Ring ◽  
Injective Modules

One of the continuity conditions identified by Utumi on self-injective rings is the C3-condition, where a module M is called a C3-module if whenever A and B are direct summands of M and A ∩ B=0, then A ⊕ B is a summand of M. In addition to injective and direct-injective modules, the class of C3-modules includes the semisimple, continuous, indecomposable and regular modules. Indeed, every commutative ring is a C3-ring. In this paper we provide a general and unified treatment of the above mentioned classes of modules in terms of the C3-condition, and establish new characterizations of several well known classes of rings.

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