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 A note on extensions of Baer and P. P. -rings

1974 ◽  
Vol 18 (4) ◽  
pp. 470-473 ◽  
Author(s):  
Efraim P. Armendariz
Keyword(s):  
Left Ideal ◽  
Polynomial Identity ◽  
Polynomial Rings ◽  
Nilpotent Elements ◽  
Baer Ring ◽  
Baer Rings ◽  
Left Annihilator

Baer rings are rings in which the left (right) annihilator of each subset is generated by an idempotent [6]. Closely related to Baer rings are left P.P.-rings; these are rings in which each principal left ideal is projective, or equivalently, rings in which the left annihilator of each element is generated by an idempotent. Both Baer and P.P.-rings have been extensively studied (e.g. [2], [1], [3], [7]) and it is known that both of these properties are not stable relative to the formation of polynomial rings [5]. However we will show that if a ring R has no nonzero nilpotent elements then R[X] is a Baer or P.P.-ring if and only if R is a Baer or P.P.-ring. This generalizes a result of S. Jøndrup [5] who proved stability for commutative P.P.-rings via localizations – a technique which is, of course, not available to us. We also consider the converse to the well-known result that the center of a Baer ring is a Baer ring [6] and show that if R has no nonzero nilpotent elements, satisfies a polynomial identity and has a Baer ring as center, then R must be a Baer ring. We include examples to illustrate that all the hypotheses are needed.

Modules Whose Endomorphism Rings Are (m, n)-Coherent

2019 ◽  
Vol 26 (02) ◽  
pp. 231-242
Author(s):  
Xiaoqiang Luo ◽  
Lixin Mao
Keyword(s):  
Left Ideal ◽  
Endomorphism Ring ◽  
Finitely Generated ◽  
Endomorphism Rings ◽  
Von Neumann ◽  
Von Neumann Regular ◽  
Coherent Ring ◽  
Left Annihilator

Let M be a right R-module with endomorphism ring S. We study the left (m, n)-coherence of S. It is shown that S is a left (m, n)-coherent ring if and only if the left annihilator [Formula: see text] is a finitely generated left ideal of Mn(S) for any M-m-generated submodule X of Mn if and only if every M-(n, m)-presented right R-module has an add M-preenvelope. As a consequence, we investigate when the endomorphism ring S is left coherent, left pseudo-coherent, left semihereditary or von Neumann regular.

scholarly journals Quasi-Baer ring extensions and biregrular rings

2000 ◽  
Vol 61 (1) ◽  
pp. 39-52 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Left Ideal ◽  
Regular Ring ◽  
Nonempty Subset ◽  
Baer Ring ◽  
Principal Ideals ◽  
Ring Extensions ◽  
Left Annihilator

A ring R with unity is called a (quasi-) Baer ring if the left annihilator of every (left ideal) nonempty subset of R is generated (as a left ideal) by an idempotent. Armendariz has shown that if R is a reduced PI-ring whose centre is Baer, then R is Baer. We generalise his result by considering the broader question: when does the (quasi-) Baer condition extend to a ring from a subring? Also it is well known that a regular ring is Baer if and only if its lattice of principal right ideals is complete. Analogously, we prove that a biregular ring is quasi-Baer if and only if its lattice of principal ideals is complete.

Endo-principally quasi-Baer modules

2015 ◽  
Vol 15 (02) ◽  
pp. 1550132 ◽  
Author(s):  
P. Amirzadeh Dana ◽  
A. Moussavi
Keyword(s):  
Direct Summand ◽  
Endomorphism Ring ◽  
Free Module ◽  
Baer Ring ◽  
Baer Rings ◽  
Baer Modules

Analogous to left p.q.-Baer property of a ring [G. F. Birkenmeier, J. Y. Kim and J. K. Park, Principally quasi-Baer rings, Comm. Algebra29 (2001) 639–660], we say a right R-module M is endo-principallyquasi-Baer (or simply, endo-p.q.-Baer) if for every m ∈ M, lS(Sm) = Se for some e2 = e ∈ S = End R(M). It is shown that every direct summand of an endo-p.q.-Baer module inherits the property that any projective (free) module over a left p.q.-Baer ring is an endo-p.q.-Baer module. In particular, the endomorphism ring of every infinitely generated free right R-module is a left (or right) p.q.-Baer ring if and only if R is quasi-Baer. Furthermore, every principally right ℱℐ-extending right ℱℐ-𝒦-nonsingular ring is left p.q.-Baer and every left p.q.-Baer right ℱℐ-𝒦-cononsingular ring is principally right ℱℐ-extending.

Baer Endomorphism Rings and Closure Operators

1978 ◽  
Vol 30 (5) ◽  
pp. 1070-1078 ◽  
Author(s):  
Soumaya M. Khuri
Keyword(s):  
Endomorphism Ring ◽  
Free Module ◽  
Linear Transformations ◽  
Endomorphism Rings ◽  
Dual Vector ◽  
Annihilator Ideal ◽  
Baer Ring ◽  
Left Annihilator ◽  
Torsion Free ◽  
Made In

A Baer ring is a ring in which every right (and left) annihilator ideal is generated by an idempotent. Generalizing quite naturally from the fact that the endomorphism ring of a vector space is a Baer ring, Wolfson [5; 6] investigated questions such as when the endomorphism ring of a free module is a Baer ring, and when the ring of continuous linear transformations on a pair of dual vector spaces is a Baer ring. A further generalization was made in [7], where the question of when the endomorphism ring of a torsion-free module over a semiprime left Goldie ring is a Baer ring was treated.

scholarly journals On Maximal Rings of Right Quotients

1962 ◽  
Vol 5 (2) ◽  
pp. 147-149 ◽  
Author(s):  
Joanne Christensen
Keyword(s):  
Left Ideal ◽  
Endomorphism Rings ◽  
Division Rings ◽  
Left Annihilator

Utumi has shown [3, Claim 5.1] that for a certain class of rings the associated maximal rings of right quotients are isomorphic to the endomorphism rings of modules over division rings. We shall prove a generalization of this theorem and then show how it is obtained as a corollary. The following proofs do not depend on Utumi's paper; instead, they make extensive use of results proved in [1]. The terminology and notations employed here are the same as in [1].I wish to thank Dr. B. Banaschewski for his suggestions and helpful criticism.LEMMA: If J is a left ideal with zero left annihilator in a ring R then a maximal ring of right quotients of R is also a maximal ring of right quotients of J.

scholarly journals Baer and quasi-Baer properties of group rings

2007 ◽  
Vol 83 (2) ◽  
pp. 285-296 ◽  
Author(s):  
Zhong Yi ◽  
Yiqiang Zhou
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Nonempty Subset ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Baer Group ◽  
Baer Ring ◽  
Fixed Ring ◽  
A Finite Group ◽  
Left Annihilator

AbstractA ring R is said to be a Baer (respectively, quasi-Baer) ring if the left annihilator of any nonempty subset (respectively, any ideal) of R is generated by an idempotent. It is first proved that for a ring R and a group G, if a group ring RG is (quasi-) Baer then so is R; if in addition G is finite then |G|–1 € R. Counter examples are then given to answer Hirano's question which asks whether the group ring RG is (quasi-) Baer if R is (quasi-) Baer and G is a finite group with |G|–1 € R. Further, efforts have been made towards answering the question of when the group ring RG of a finite group G is (quasi-) Baer, and various (quasi-) Baer group rings are identified. For the case where G is a group acting on R as automorphisms, some sufficient conditions are given for the fixed ring RG to be Baer.

π-Baer rings

2018 ◽  
Vol 17 (02) ◽  
pp. 1850029 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Yeliz Kara ◽  
Adnan Tercan
Keyword(s):  
Left Ideal ◽  
Matrix Ring ◽  
Idempotent Element ◽  
Matrix Rings ◽  
Full Matrix ◽  
Baer Ring ◽  
Baer Rings ◽  
Base Ring ◽  
Full Matrix Ring ◽  
The Right

We say a ring [Formula: see text] is [Formula: see text]-Baer if the right annihilator of every projection invariant left ideal of [Formula: see text] is generated by an idempotent element of [Formula: see text]. In this paper, we study connections between the [Formula: see text]-Baer condition and related conditions such as the Baer, quasi-Baer and [Formula: see text]-extending conditions. The [Formula: see text]-by-[Formula: see text] generalized triangular and the [Formula: see text]-by-[Formula: see text] triangular [Formula: see text]-Baer matrix rings are characterized. Also, we prove that a [Formula: see text]-by-[Formula: see text] full matrix ring over a [Formula: see text]-Baer ring is a [Formula: see text]-Baer ring. In contrast to the Baer condition, it is shown that the [Formula: see text]-Baer condition transfers from a base ring to many of its polynomial extensions. Examples are provided to illustrate and delimit our results.

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.

Partial orders on the power sets of Baer rings

2019 ◽  
Vol 19 (01) ◽  
pp. 2050011 ◽  
Author(s):  
B. Ungor ◽  
S. Halicioglu ◽  
A. Harmanci ◽  
J. Marovt
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Linear Operators ◽  
Bounded Linear ◽  
Von Neumann ◽  
Baer Ring ◽  
Power Set ◽  
Von Neumann Regular ◽  
Baer Rings ◽  
The Right

Let [Formula: see text] be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set [Formula: see text] of [Formula: see text] and show that this relation, which we call “the minus order on [Formula: see text]”, is a partial order when [Formula: see text] is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer ∗-rings. We show that some ideals generated by projections of a von Neumann regular and Baer ∗-ring [Formula: see text] form a lattice with respect to the star partial order on [Formula: see text]. As a particular case, we present characterizations of these orders on the power set of [Formula: see text], the algebra of all bounded linear operators on a Hilbert space [Formula: see text].

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