Purities and Pure Injective Modules over Serial, Right Noetherian Rings

2004 ◽  
Vol 124 (1) ◽  
pp. 4796-4805
Author(s):  
I. M. Zilberbord
Keyword(s):  
Noetherian Rings ◽  
Injective Modules ◽  
Right Noetherian Rings

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.

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].

3.—A Note on Projective Modules

1976 ◽  
Vol 75 (1) ◽  
pp. 24-31 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Noetherian Rings ◽  
Projective Modules ◽  
Finitely Generated ◽  
Right Noetherian Rings

SynopsisFor various classes of right noetherian rings it is shown that projective right modules are either finitely generated or free.

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.

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.

Completion of Modules over Serial, Right Noetherian Rings

2004 ◽  
Vol 120 (4) ◽  
pp. 1583-1590
Author(s):  
A. I. Generalov ◽  
I. M. Zilberbord
Keyword(s):  
Noetherian Rings ◽  
Right Noetherian Rings

scholarly journals INJECTIVE MODULES OVER ω-NOETHERIAN RINGS, II

2013 ◽  
Vol 50 (5) ◽  
pp. 1051-1066 ◽  
Author(s):  
Jun Zhang ◽  
Fanggui Wang ◽  
Hwankoo Kim
Keyword(s):  
Noetherian Rings ◽  
Injective Modules

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 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.

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