Endomorphism Rings of Abelian Groups

Крылов, П. А.; Mikhalëv, A. V.; Tuganbaev, A. A.

doi:10.1007/978-94-017-0345-1

Cited by 108 publications

(84 citation statements)

References 207 publications

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“…6 and [38,40]). The module with discrete endomorphism ring is self-small [38]. We define the category Walk of Walker (see [40]).…”

Section: Idempotent Functors and Localizations In The Category Of Abementioning

confidence: 94%

“…The functors Hom P (A, −) and (−) ⊗ S A define an equivalence between the category of A-soluble modules and the category M A (see [38,Sec. 34]).…”

Section: Localizations Of Some Modules and Ringsmentioning

confidence: 99%

“…is an isomorphism of P -modules; the term "an A-soluble module" is used in [38]. Basically, we consider either A-soluble modules X, or modules X such that the trace tr A X of the module A in X is an A-soluble module (see the definition of the trace in Sec.…”

Section: Localizations Of Some Modules and Ringsmentioning

confidence: 99%

“…For example, they are successively used in the study of torsion-free groups and their endomorphism rings (see [38]). The role of E-rings was later observed in homotopy theory.…”

mentioning

confidence: 99%

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Idempotent functors and localizations in categories of modules and Abelian groups

Крылов

Tuganbaev²

2012

J Math Sci

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“…6 and [38,40]). The module with discrete endomorphism ring is self-small [38]. We define the category Walk of Walker (see [40]).…”

Section: Idempotent Functors and Localizations In The Category Of Abementioning

confidence: 94%

“…The functors Hom P (A, −) and (−) ⊗ S A define an equivalence between the category of A-soluble modules and the category M A (see [38,Sec. 34]).…”

Section: Localizations Of Some Modules and Ringsmentioning

confidence: 99%

Section: Localizations Of Some Modules and Ringsmentioning

confidence: 99%

“…For example, they are successively used in the study of torsion-free groups and their endomorphism rings (see [38]). The role of E-rings was later observed in homotopy theory.…”

mentioning

confidence: 99%

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Idempotent functors and localizations in categories of modules and Abelian groups

Крылов

Tuganbaev²

2012

J Math Sci

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“…Generalized matrix rings appear in the theory of rings and modules in quite various situations. Such rings, for instance, have found interesting application in treating endomorphism rings of Abelian groups [9][10][11]. Triangular generalized matrix rings, as is well known, are a source for constructing rings with asymmetric properties.…”

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confidence: 98%

Isomorphism of generalized matrix rings

Крылов¹

2008

Algebra Logic

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An isomorphism problem is considered for generalized matrix rings with values in a given ring R. An exhaustive answer is given for the case of a commutative domain R and a commutative local ring R.For generalized (or formal) matrix rings and their connections with Morita contexts, we ask the reader to consult [1(Chap. 12)-3]. These rings have been intensively studied as of late (see, e.g., [4][5][6][7][8]). Generalized matrix rings appear in the theory of rings and modules in quite various situations. Such rings, for instance, have found interesting application in treating endomorphism rings of Abelian groups [9][10][11]. Triangular generalized matrix rings, as is well known, are a source for constructing rings with asymmetric properties. A chapter in [2] is concerned with just these rings.A generalized matrix ring is defined by two bimodule homomorphisms. The choice of another pair of homomorphisms generally leads up to another ring. There arises a problem of classifying generalized matrix rings in relation to corresponding pairs of bimodule homomorphisms. In the present paper we solve this problem for generalized matrix rings with values in a given ring R. The answer is obtained for the case of a commutative domain R and a commutative local ring R.We give the definition of a generalized matrix ring. Let R and S be rings, M an R-S-bimodule, and N an S-R-bimodule. Assume that there are bimodule homomorphisms ϕ : M ⊗ S N → R and ψ : N ⊗ R M → S satisfying the following associativity conditions: (mn)m = m(nm ) and (nm)n = n(mn ) for all m, m ∈ M and n, n ∈ N . Put mn = ϕ(m ⊗ n) and nm = ψ(n ⊗ m). Consider a set R M N S of all matrices r m n s , where r ∈ R, s ∈ S, m ∈ M , and n ∈ N . With respect to operations specified as in the usual matrix ring, this set is a ring, which is called a generalized matrix ring (of order 2). We fix the letter K to denote the introduced ring R M N S . If emphasis should be laid on the fact that K is constructed using homomorphisms ϕ and ψ, then we write K(ϕ, ψ). A ring K(ϕ, ψ) with zero homomorphisms ϕ and ψ is said to be trivial. Triangular generalized matrix rings pertain to trivial ones (if M = 0 or N = 0). In a similar way, we can define a generalized matrix ring of any order n > 2. Such a ring is naturally isomorphic to a generalized matrix ring of order 2 (the isomorphism is obtained by partitioning every matrix into four blocks). Usually, from technical considerations, order 2 generalized matrix rings are considered. We do so in the present paper, too. A class of generalized matrix rings coincides with a class pr. Mira 3-60, Tomsk, 634057 Russia; krylov@math.tsu.ru.

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Transformation Near-Rings

Lockhart¹

2021

The Theory of Near-Rings

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Endomorphism Rings of Abelian Groups

Cited by 108 publications

References 207 publications

Idempotent functors and localizations in categories of modules and Abelian groups

Idempotent functors and localizations in categories of modules and Abelian groups

Isomorphism of generalized matrix rings

Transformation Near-Rings

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