The Maximal Regular Ideal of a Ring

Brown, Bailey; McCoy, Neal H.

doi:10.2307/2031919

Cited by 19 publications

(29 citation statements)

References 2 publications

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“…This submodule Reg M is obtained by applying the isomorphism (1). Furthermore, for M = R we obtain the maximum regular ideal M R of Brown and McCoy (1950)…”

Section: The Existence Of Reg a Mmentioning

confidence: 99%

Regularity and Substructures of Hom

Kasch

Mader

2006

Communications in Algebra

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We define "regular" for maps in a Hom group. This notion specializes to the well-known notions of Von Neumann regular in rings and modules. A map f ∈ Hom R A M is regular if and only if Ker f ⊆ ⊕ A and Im f ⊆ ⊕ M. There exists a unique maximal regular End M -End A -submodule in Hom R A M . We study regularity in Hom R A 1 ⊕ A 2 M 1 ⊕ M 2 . The existence of a regular function Hom R A M implies the existence of projective summands of Hom R A M End R A and of End R M Hom R A M . We consider regularity in endomorphism rings, and generalize a theorem of Ware-Zelmanowitz. We examine connections between the maximum regular bimodule and other substructures of Hom, mention two generalizations of regularity, and raise some questions.

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“…This submodule Reg M is obtained by applying the isomorphism (1). Furthermore, for M = R we obtain the maximum regular ideal M R of Brown and McCoy (1950)…”

Section: The Existence Of Reg a Mmentioning

confidence: 99%

Regularity and Substructures of Hom

Kasch

Mader

2006

Communications in Algebra

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“…In their 1950 paper [4], Brown and McCoy proved that every non-unital ring R has a unique maximal regular two-sided ideal M(R) with the following "radical-like" properties: [12]. Here we prove some further properties and answer a question raised in [12].…”

Section: A Results On Regularitymentioning

confidence: 59%

“…(3) There exists a regular element c ∈ R such that a − c ∈ J (R). (4) There exists b ∈ R with bab = b and a − aba ∈ J (R).…”

Section: The Maximal Semiregular Ideal Of a Ringmentioning

confidence: 99%

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On (semi)regularity and the total of rings and modules

Zhou

2009

Journal of Algebra

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Let M and N be two modules over a ring R. Recent works by Kasch, Schneider, Beidar, Mader, and others have shown that some of the ring and module theory can be carried out in the context of hom R (M, N). The study of substructures of hom R (M, N) such as the radical, the singular and co-singular ideals and the total has raised new questions for research in this area. This paper is a continuation of study of these substructures, focusing on when the total is equal to the radical, as well as their connections with (semi)regularity of hom R (M, N). New results obtained include necessary and sufficient conditions for the total to equal the radical, a description of the maximal regular sub-bimodule of hom R (M, N), the existence of the maximal semiregular ideal of a ring, and answers to a number of existing open questions.

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“…Let R satisfy the above conditions, and let M be the maximal regular ideal of R. Since MC= R 2 in any ring, and R 2 is itself regular, it follows that M= R 2. By Theorem 7, [1], R~R2@B…”

Section: ) Every Bitranslation On R Is Induced By At Most One Elementmentioning

confidence: 97%

On the ring of bitranslations of certain rings

Petrich

1969

Acta Mathematica Academiae Scientiarum Hungaricae

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1. Introduction. In the theory of ring extensions, the notion of a bitranslation (our terminology, for the definition see below) plays the role of automorphisms in group extensions. Bitranslations were studied by HOCHSCmLD [5] for algebras (,,multiplications", they aye also linear transformations); by CLIFFORD [2] for semigroups (,,pairs of linked left and right translations", where only the multiplicative part of the definition is retained); by RfiDEI [9] (,,Doppelhomothetismen") and MAC LANE [7] for rings (,,bimultiplications"), among others.Under a natural addition and multiplication, the set of bitranslations of a ring itself forms a ring. RfiDEI [9] defines a holomorph of a ring R as a (natural) split extension of R by a maximal ring of permutable bitranslations. It turns out that R may have more than one holomorph but that otherwise these have several properties analogous to the properties of the holomorph of a group. A number of properties of a ring holomorph have been established by RfiDE~ [8], [9], SZENDREI [10], VAN LEEUWEN [6], and WEINERT and EILHAUER [11].The purpose of this paper is to study the ring Y2(R) of bitranslations of a ring R for which czfl=fl~=0 for all flCR implies a=0 (see [5], [7], [11]). In this case Y2(R) is a ring of permutable bitranslations and the split extension of R by O(R) is the unique holomorph of R. The analogy with the group holomorph is in this case much stronger than in the case of an arbitrary ring. Section 2 contains most of the necessary definitions and notation. We are mainly concerned with rings R satisfying the above condition; for such R, in Section 3, we establish several properties of Q(R), and in Section 4, characterize s~(R) in several ways. We conclude in Section 5 by constructing Q(R) for a ring R which is of a special kind but need not satisfy the above condition, and derive several consequences of this result.It is of interest that the theory of ideal extensions of semigroups is quite similar to the theory of ring extensions, at least as far as multiplication is concerned (see [2] and 4.4, [3]). Even though the multiplicative part of a ring is a semigroup, this still seems surprising in view of very different definitions of extensions in semigroups and rings.From the results of Sections 3 and 4, we see that for a ring R satisfying the above condition, g2(R) plays the role of a ,,holomorph" of R (cf. ~ 54, [8]).

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The Maximal Regular Ideal of a Ring

Cited by 19 publications

References 2 publications

Regularity and Substructures of Hom

Regularity and Substructures of Hom

On (semi)regularity and the total of rings and modules

On the ring of bitranslations of certain rings

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