Prime and coprime modules

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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“…In this section, we address this question. Notation, terminology, and basic results can be found in [Ste75] and [BKN82].…”

Section: Torsion Theorymentioning

confidence: 99%

s.Baer and s.Rickart Modules

Birkenmeier

Leblanc²

2015

J. Algebra Appl.

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“…al. [BJKN80] to present the notion of coprime modules. The definition was modified in [RRW05], where arbitrary submodules are replaced by the fully invariant ones.…”

Section: Fully Coprime (Fully Cosemiprime) Comodulesmentioning

confidence: 99%

Fully Coprime Comodules and Fully Coprime Corings

Abuhlail

2006

Appl Categor Struct

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Prime objects were defined as generalization of simple objects in the categories of rings (modules). In this paper we introduce and investigate what turns out to be a suitable generalization of simple corings (simple comodules), namely fully coprime corings (fully coprime comodules). Moreover, we consider several primeness notions in the category of comodules of a given coring and investigate their relations with the fully coprimeness and the simplicity of these comodules. These notions are applied then to study primeness and coprimeness properties of a given coring, considered as an object in its category of right (left) comodules.

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“…In [2], a module M R is called prime if Hom R (M, K)N ̸ = 0 for all nonzero submodules K, N ≤ M R and it is shown that M R is prime if and only if it is cogenerated by each of its nonzero submodules. A semiprime notion for modules is then obtained in [4] by setting K = N in the above definition of prime modules.…”

Section: Introductionmentioning

confidence: 99%

“…We also call M R semiprime if every essential submodule of M R cogenerates M R . In this paper, prime module means the prime module in the sense of [2]; see [11,Sections 13,14] for an excellent reference on the subject. Weakly compressible modules have applied in different situations.…”

Section: Introductionmentioning

confidence: 99%

Semiprime and Weakly Compressible Modules

Dehghani¹,

Vedadi²

2016

HJMS

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An R-module M is called semiprime (resp. weakly compressible) if it is cogenerated by each of its essential submodules (resp. HomR(M, N )N is nonzero for every 0 ̸ = N ≤ MR). We carry out a study of weakly compressible (semiprime) modules and show that there exist semiprime modules which are not weakly compressible. Weakly compressible modules with enough critical submodules are characterized in different ways. For certain rings R, including prime hereditary Noetherian rings, it is proved that MR is weakly compressible (resp. semiprime) if and

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Prime and coprime modules

Cited by 51 publications

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s.Baer and s.Rickart Modules

s.Baer and s.Rickart Modules

Fully Coprime Comodules and Fully Coprime Corings

Semiprime and Weakly Compressible Modules

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