Extensions of Rings and Modules

Birkenmeier, Gary F.; Park, Jae Keol; Rizvi, S. Tariq

doi:10.1007/978-0-387-92716-9

Cited by 96 publications

(54 citation statements)

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“…endowed with the usual matrix operations is an R-algebra called a triangular matrix R-algebra (see [5] and [6] for more details about this construction). Following [5,Chapter 5], an algebra S is said to have a triangular matrix representation if S is isomorphic to a triangular matrix algebra. By [5, Theorem 5.1.4], a unital algebra A has a triangular matrix representation if there exists a non trivial idempotent e ∈ A such that (1 − e)Ae = 0.…”

Section: Trivial Extension Algebras and Triangular Matrix Algebrasmentioning

confidence: 99%

Derivations and the First Cohomology Group of Trivial Extension Algebras

Bennis

Fahid

2017

Mediterr. J. Math.

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Abstract. In this paper we investigate in details derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the trivial extension algebras and the triangular matrix algebras.

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Section: Trivial Extension Algebras and Triangular Matrix Algebrasmentioning

confidence: 99%

Derivations and the First Cohomology Group of Trivial Extension Algebras

Bennis

Fahid

2017

Mediterr. J. Math.

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“…equipped with the usual matrix operations is an R-algebra called a (generalized) triangular R-algebra (see [10] for more details about this construction). In this paper we assume that M is also a faithful (A, B)-bimodule.…”

Section: Introductionmentioning

confidence: 99%

f-Biderivations and Jordan biderivations of unital algebras with idempotents

et al. 2020

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The notion of f -derivations was introduced by Beidar and Fong to unify several kinds of linear maps including derivations, Lie derivations and Jordan derivations. In this paper we introduce the notion of f -biderivations as a natural "biderivation" counterpart of the notion of "f -derivations". We first show, under some conditions, that any f -biderivation is a Jordan biderivation. Then, we turn to study f -biderivations of a unital algebra with an idempotent. Our second main result shows, under some conditions, that every Jordan biderivation can be written as a sum of a biderivation, an antibiderivation and an extremal biderivation. As a consequence we show that every Jordan biderivation on a triangular algebra is a biderivation.

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“…Baer and Rickart rings have a long history dating back to the 1940s with roots in functional analysis. For more on these topics see [Ber], [BPR13], [End60], [Goo80], [Hat60], [Kap68], and [Ric46].…”

Section: Introductionmentioning

confidence: 99%

“…The following result from [San67, Proposition 1.2] will be used implicitly throughout this paper: for X, Y ∈ Mod R if X ≤ ess Y , then for all y ∈ Y , y −1 X = {r ∈ R | yr ∈ X} ≤ ess R R . Lastly, undefined notation or terminology can be found in [BLM04], [BPR13], [Kap53], and [Lam99].…”

Section: Introductionmentioning

confidence: 99%

s.Baer and s.Rickart Modules

Birkenmeier

Leblanc²

2015

J. Algebra Appl.

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Abstract. In this paper, we study module theoretic definitions of the Baer and related ring concepts. We say a module is s.Baer if the right annihilator of a nonempty subset of the module is generated by an idempotent in the ring. We show that s.Baer modules satisfy a number of closure properties. Under certain conditions, a torsion theory is established for the s.Baer modules, and we provide examples of s.Baer torsion modules and modules with a nonzero s.Baer radical. The other principal interest of this paper is to provide explicit connections between s.Baer modules and projective modules. Among other results, we show that every s.Baer module is an essential extension of a projective module. Additionally, we prove, with limited and natural assumptions, that in a generalized triangular matrix ring every s.Baer submodule of the ring is projective. As an application, we show that every prime ring with a minimal right ideal has the strong summand intersection property. Numerous examples are provided to illustrate, motivate, and delimit the theory. IntroductionA Baer ring is a ring in which the right annihilator of an arbitrary nonempty subset is generated by an idempotent. A more general notion of a Baer ring is that of a right Rickart ring where the right annihilator of an arbitrary element is generated by an idempotent. A ring is right Rickart if and only if every principal right ideal is projective. Hence these rings are often referred to as right p.p. rings. Baer and Rickart rings have a long history dating back to the 1940s with roots in functional analysis. For more on these topics see [Ber] The notions of a Baer and Rickart module that we shall consider in this paper are exactly the definitions used by Evans, Lee and Zhou, and Liu and Chen. Thus, a module M R is called s.Rickart if, for any m ∈ M R , r R (m) = eR for some e = e 2 ∈ R. A module M R is called s.Baer if, for any nonempty subset S of M , r R (S) = eR for e = e 2 ∈ R. To contrast, we denote the Baer module concept of [RR04] by e.Baer. Note that when M R = R R all the aforementioned notions of a Baer module coincide.In Section 1, we investigate a number of closure properties for s.Baer modules: submodules, direct sums, direct products, and module extensions. When R has the SSIP or is orthogonally finite, the classes of s.Baer and s.Rickart modules coincide and are closed under direct products. We determine conditions on (s.Baer) s.Rickart modules which ensure that R has the (S)SIP. For a simple module M , M is nonsingular ⇐⇒ projective ⇐⇒ s.Rickart ⇐⇒ s.Baer. Then we characterize the primitive rings which have a faithful simple s.Baer module. Surprisingly, we prove that a right primitive ring with nonzero socle has the SSIP. If M is s.Baer, we show when Hom(M, −) and Hom(−, M ) are s.Baer. A ring R is semisimple Artinian if and only if every R-module is s.Baer. Finally, we discuss conditions on R such that all nonsingular modules are s.Baer.In Section 2, we begin exploring connections with projectivity. In particular, every s.Rickart module is an...

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Extensions of Rings and Modules

Cited by 96 publications

References 0 publications

Derivations and the First Cohomology Group of Trivial Extension Algebras

Derivations and the First Cohomology Group of Trivial Extension Algebras

f-Biderivations and Jordan biderivations of unital algebras with idempotents

s.Baer and s.Rickart Modules

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