Semiregular Modules and Rings

1976 ◽  
Vol 28 (5) ◽  
pp. 1105-1120 ◽  
Author(s):  
W. K. Nicholson
Keyword(s):  
Direct Summand ◽  
Homomorphic Image ◽  
Projective Module ◽  
Structure Theorem ◽  
Projective Cover ◽  
Finitely Generated

Mares [9] has called a projective module semiperfect if every homomorphic image has a projective cover and has shown that many of the properties of semiperfect rings can be extended to these modules. More recently Zelmanowitz [16] has called a module regular if every finitely generated submodule is a projective direct summand. In the present paper a class of semiregular modules is introduced which contains all regular and all semiperfect modules. Several characterizations of these modules are given and a structure theorem is proved. In addition several theorems about regular and semiperfect modules are extended.

scholarly journals On Quasi-h-Dense Submodules and h-Pure Envelopes of QTAG Modules

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Alveera Mehdi ◽  
Fahad Sikander ◽  
Firdhousi Begum
Keyword(s):  
Direct Summand ◽  
Homomorphic Image ◽  
Associative Ring ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules ◽  
Pure Submodules

A module M over an associative ring R with unity is a QTAG module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. There are many fascinating properties of QTAG modules of which h-pure submodules and high submodules are significant. A submodule N is quasi-h-dense in M if M/K is h-divisible, for every h-pure submodule K of M, containing N. Here we study these submodules and obtain some interesting results. Motivated by h-neat envelope, we also define h-pure envelope of a submodule N as the h-pure submodule K⊇N if K has no direct summand containing N. We find that h-pure envelopes of N have isomorphic basic submodules, and if M is the direct sum of uniserial modules, then all h-pure envelopes of N are isomorphic.

On Semiperfect Modules

1975 ◽  
Vol 18 (1) ◽  
pp. 77-80 ◽  
Author(s):  
W. K. Nicholson
Keyword(s):  
Homomorphic Image ◽  
Projective Module ◽  
Simple Module ◽  
Projective Cover ◽  
Maximal Submodule

AbstractSandomierski (Proc. A.M.S. 21 (1969), 205–207) has proved that a ring is semiperfect if and only if every simple module has a projective cover. This is generalized to semiperfect modules as follows: If P is a projective module then P is semiperfect if and only if every simple homomorphic image of P has a projective cover and every proper submodule of P is contained in a maximal submodule.

Inverse semigroups generated by a pair of subgroups

1977 ◽  
Vol 77 (1-2) ◽  
pp. 9-22 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Homomorphic Image ◽  
Main Part ◽  
Structure Theorem ◽  
Inverse Semigroups ◽  
Infinite Cyclic Group ◽  
The Third ◽  
Group A ◽  
Maximum Group

SynopsisThe aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.

scholarly journals When every projective module is a direct sum of finitely generated modules

2007 ◽  
Vol 315 (1) ◽  
pp. 454-481 ◽  
Author(s):  
Warren Wm. McGovern ◽  
Gena Puninski ◽  
Philipp Rothmaler
Keyword(s):  
Projective Module ◽  
Finitely Generated ◽  
Direct Sum ◽  
Finitely Generated Modules

On the cofiniteness of Artinian local cohomology modules

2016 ◽  
Vol 15 (04) ◽  
pp. 1650070 ◽  
Author(s):  
Ghader Ghasemi ◽  
Kamal Bahmanpour ◽  
Jafar A’zami
Keyword(s):  
Local Ring ◽  
Homomorphic Image ◽  
Local Cohomology ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Gorenstein Local Ring ◽  
Cohomology Modules

Let [Formula: see text] be a commutative Noetherian local ring, which is a homomorphic image of a Gorenstein local ring and [Formula: see text] an ideal of [Formula: see text]. Let [Formula: see text] be a nonzero finitely generated [Formula: see text]-module and [Formula: see text] be an integer. In this paper we show that, the [Formula: see text]-module [Formula: see text] is nonzero and [Formula: see text]-cofinite if and only if [Formula: see text]. Also, several applications of this result will be included.

scholarly journals Rings characterised by semiprimitive modules

1995 ◽  
Vol 52 (1) ◽  
pp. 107-116
Author(s):  
Yasuyuki Hirano ◽  
Dinh Van Huynh ◽  
Jae Keol Park
Keyword(s):  
Direct Summand ◽  
Projective Module ◽  
Injective Module ◽  
Direct Sum

A module M is called a CS-module if every submodule of M is essential in a direct summand of M. It is shown that a ring R is semilocal if and only if every semiprimitive right R-module is CS. Furthermore, it is also shown that the following statements are equivalent for a ring R: (i) R is semiprimary and every right (or left) R-module is injective; (ii) every countably generated semiprimitive right R-module is a direct sum of a projective module and an injective module.

BAER ENDOMORPHISM RINGS AND ENVELOPES

2010 ◽  
Vol 09 (03) ◽  
pp. 365-381 ◽  
Author(s):  
LIXIN MAO
Keyword(s):  
Direct Summand ◽  
Left Ideal ◽  
Nonempty Subset ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Baer Ring ◽  
Baer Rings ◽  
Left Annihilator

R is called a Baer ring if the left annihilator of every nonempty subset of R is a direct summand of RR. R is said to be a left AFG ring in case the left annihilator of every nonempty subset of R is a finitely generated left ideal. In this paper, we study Baer rings and AFG rings of endomorphisms of modules in terms of envelopes. Some known results are extended.

scholarly journals Nice Bases of QTAG-Modules

2021 ◽  
pp. 1-5
Author(s):  
Fahad Sikander ◽  
Tanveer Fatima ◽  
Ayazul Hasan
Keyword(s):  
Homomorphic Image ◽  
Associative Ring ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules ◽  
Divisible Hull

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of universal modules. In this paper, we investigate the class of QTAG-modules having nice basis. It is proved that if H_ω (M) is bounded then M has a bounded nice basis and if H_ω (M) is a direct sum of uniserial modules, then M has a nice basis. We also proved that if M is any QTAG-module, then M⊕D has a nice basis, where D is the h-divisible hull of H_ω (M).

The Maximal Co-Rational Extension by a Module

1966 ◽  
Vol 18 ◽  
pp. 953-962 ◽  
Author(s):  
R. C. Courter
Keyword(s):  
Projective Module ◽  
Unit Element ◽  
Projective Cover ◽  
Arbitrary Ring ◽  
Rational Extension ◽  
Image Position

Modules are S-modules where S is an arbitrary ring with or without a unit element. We consider a projective module P having a submodule K such that K + Y = P implies that the submodule Y is P (P, then, is a projective cover of P/K (Definition 4 in this section)) and we define the submodule X of P byOur main result states that up to isomorphism P/X is the maximal co-rational extension over P/K (by P/K, in the more precise wording of the title).

scholarly journals Convergent Sequences in Discrete Groups

2013 ◽  
Vol 56 (2) ◽  
pp. 424-433 ◽  
Author(s):  
Andreas Thom
Keyword(s):  
Homomorphic Image ◽  
Discrete Groups ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Convergent Sequences

AbstractWe prove that a finitely generated group contains a sequence of non-trivial elements that converge to the identity in every compact homomorphic image if and only if the group is not virtually abelian. As a consequence of the methods used, we show that a finitely generated group satisfies Chu duality if and only if it is virtually abelian.

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