scholarly journals WEAK and CO-WEAK BAER MODULES: موديلات بيير الضعيفة والضعيفة المرافقة

2021 ◽  
Vol 5 (4) ◽  
pp. 110-98
Author(s):  
Eaman Al-Khouja, Magd Alfakhory, Hamza Hakmi Eaman Al-Khouja, Magd Alfakhory, Hamza Hakmi
Keyword(s):  
Endomorphism Ring ◽  
Free Module ◽  
Orthogonal Idempotent ◽  
Baer Modules ◽  
Infinite Set

The object of this paper is study the notions of weak Baer and weak Rickart rings and modules. We obtained many characterizations of weak Rickart rings and provide their properties. Relations ship between a weak Rickart (weak Baer) module and its endomorphism ring are studied. We proved that a weak Baer module with no infinite set of nonzero orthogonal idempotent elements in its endomorphism ring is precisely a Baer module. In addition, the endomorphism ring of a semi-projective weak Rickart module is semi-potent and the endomorphism ring of a semi-injective coweak Rickart module is semi-potent. Furthermore, we show that a free module is weak Baer if and only if its endomorphism ring is left weak Baer.

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.

scholarly journals The characterization problem for endomorphism rings

1991 ◽  
Vol 50 (1) ◽  
pp. 116-137 ◽  
Author(s):  
J. L. García
Keyword(s):  
Endomorphism Ring ◽  
Free Module ◽  
Endomorphism Rings ◽  
Characterization Problem ◽  
Algebraic Properties

AbstractWe consider the problem of characterizing by abstract properties the rings which are isomorphic to the endomorphism ring End (RF) of some free module F over a ring R in a given class R of rings. We solve this problem when R is any class of rings (by employing topological notions) and when R is the class of all the left Kasch rings (in terms of algebraic properties only).

Isomorphisms of Prime Goldie Semi-Principal Left Ideal Rings, II

1988 ◽  
Vol 31 (3) ◽  
pp. 374-379 ◽  
Author(s):  
Kenneth G. Wolfson
Keyword(s):  
Left Ideal ◽  
Endomorphism Ring ◽  
Finite Rank ◽  
Sufficient Conditions ◽  
Free Module ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Ore Domain ◽  
Necessary And Sufficient

AbstractA prime Goldie ring K, in which each finitely generated left ideal is principal is the endomorphism ring E(F, A) of a free module A, of finite rank, over an Ore domain F. We determine necessary and sufficient conditions to insure that whenever K ≅ E(F, A) ≅ E(G, B) (with A and B free and finitely generated over domains F and G) then (F, A) is semi-linearly isomorphic to (G, B). We also show, by example, that it is possible for K ≅ E(F, A ) ≅ E(G, B), with F and G, not isomorphic.

scholarly journals On Frobenius Extensions I.

1960 ◽  
Vol 17 ◽  
pp. 89-110 ◽  
Author(s):  
Tadasi Nakayama ◽  
Tosiro Tsuzuku
Keyword(s):  
Commutative Ring ◽  
Endomorphism Ring ◽  
Projective Module ◽  
Commutative Rings ◽  
Free Module ◽  
General Tendency ◽  
General Notion ◽  
Frobenius Algebras ◽  
Definition Of

As a generalization of the notion of Frobenius algebras over a field Kasch [103 introduced that of Frobenius extensions of a ring. The present writers [13] recently freed one of Kasch’s main theorems from its rather strong S-ring assumption of the ground ring. However, even with the removal of the S-ring assumption of the ground ring the notion does not seem general enough, and we wish, in the present paper and its sequel, to develope the theory upon the basis of a more general notion of Frobenius extensions. Thus, we replace the free module property of the extension by the projective module property (according to a general tendency in algebra), which has been done in fact in case of Frobenius algebras over a commutative ring in a previous work by Eilenberg and one of the writers [4], and, further, take automorphisms of the ground ring into the definition of Frobenius extensions (which seems quite natural particularly in case of non-commutative rings). To such generalized notion of Frobenius extensions we may extend many of Kasch’s theorems, including those which are immediate extensions of classical theorems for Frobenius algebras and those which are essentially new, as the above alluded endomorphism ring theorem. Also homological properties of Frobenius extensions, as were developed in Hirata’s [6] recent paper in succession to Eilenberg-Nakayama [4], can be extended to our present generalized case; we shall also exceed [4], [6] somewhat in considering injective and weak dimensions.

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.

On Simple Rings with Maximal Annihilator Right Ideals

1965 ◽  
Vol 8 (5) ◽  
pp. 667-668 ◽  
Author(s):  
Kwangil Koh
Keyword(s):  
Endomorphism Ring ◽  
Integral Domain ◽  
Free Module ◽  
Simple Ring ◽  
Torsion Free Module ◽  
Torsion Free

If R is a simple ring with 1 which contains a maximal annihilator right ideal then R is the endomorphism ring of a unital torsion-free module over an integral domain.We first prove the following:Let R be a ring with 1. If a ϵ R such that (a)r = {r ϵ R|ar = 0} is a maximal annihilator right ideal then HomR(aR, aR) is an integral domain.

Endoprime modules and their direct sums

2018 ◽  
Vol 17 (08) ◽  
pp. 1850155 ◽  
Author(s):  
Gangyong Lee ◽  
S. Tariq Rizvi
Keyword(s):  
Direct Summand ◽  
Prime Ring ◽  
Endomorphism Ring ◽  
Matrix Ring ◽  
Direct Sums ◽  
Direct Sum ◽  
Finite Matrix ◽  
Reduced Ring ◽  
Baer Modules ◽  
Special Classes

The purpose of this paper is to further study the endoprime modules as one of the special classes of quasi-Baer modules. As a module theoretic analogue of a prime ring, we characterize an endoprime module via its endomorphism ring and a weak retractability condition. It is shown that any direct summand of an endoprime module is an endoprime module. A characterization is obtained when a direct sum of endoprime modules is an endoprime module. It is well known that every prime ring is semicentral reduced. We prove that a column (and row) finite matrix ring over a semicentral reduced ring is also a semicentral reduced ring. Consequently, it is shown that a column (and row) finite matrix ring over a prime ring is prime. Applications and examples illustrating our results are provided.

scholarly journals The endomorphism ring of a locally free module

1983 ◽  
Vol 35 (3) ◽  
pp. 308-326 ◽  
Author(s):  
W. N. Franzsen ◽  
P. Schultz
Keyword(s):  
Large Class ◽  
Endomorphism Ring ◽  
Special Class ◽  
Free Module ◽  
Endomorphism Rings ◽  
Free Modules

AbstractWe identify a large class of rings over which locally free modules are determined by their endomorphism rings. We characterize these endomorphism rings and consider under what circumstances the conditions on the locally free modules can be relaxed, for example by requiring that only one of the rings need be in the special class, or by replacing ‘free' by “projective”.

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