Endomorphism rings of modules and lattices of submodules

Markov, V. T.; Mikhalëv, A. V.; Skornyakov, L. A.; Tuganbaev, A. A.

doi:10.1007/bf02106808

Cited by 10 publications

(2 citation statements)

References 387 publications

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“…We give also a list of recent surveys: Gobel [98], Hahn [121], Isaacs [126], Lam [154], Marcus [176,177], Markov, Mikhalev, Skornyakov, andTuganbaev [180], May [182], Mikhalev [201], Mikhalev and Mishina [204], O'Meara [218,220], Sebeldin [249], Zalesskii [297].…”

Section: Books and Surveysmentioning

confidence: 99%

Isomorphisms and anti-isomorphisms of endomorphism rings of modules

Mikhalev¹

1996

First International Tainan-Moscow Algebra Workshop

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Let A and B be rings and Ma and Ng a right A-module and a right 5-module respectively. Let a:End(M,4) -»• End(Ar fl ) be either an isomorphism or an anti-isomorphism of their endomorphism rings. The Wedderburn-Baer-Kaplansky problem is to characterize all such a. The problem has a long history. The aim of these lectures is to consider main results in this direction for modules Ma and Ng which are close to free ones (in the epicenter we put strong generators, i.e., modules with a cyclic free direct summand). The problems under consideration are connected with conditions for inducing of isomorphisms of endomorphism rings by semilinear transformations, by Morita equivalences or by full embeddings. We consider also multiplicative isomorphisms of endomorphism semigroups of modules. We touch some related topics. First of all, we describe recent results on isomorphisms of general linear groups over arbitrary associative rings (Schreier -van der Waerden problem), because of a nice reduction of this group problem to the ring problem of classification of isomorphisms and anti-isomorphisms of matrix rings. We consider Malcev type theorems on elementary equivalences of linear groups and rings of matrices. At the same time we deal with the characterization problem of endomorphism rings of modules from different classes (in particular, of matrix rings). Some comments are related to operator rings and semigroups of topological and ordered linear spaces and to Frobenius type results on A-linear mappings on matrix rings A n .

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Section: Books and Surveysmentioning

confidence: 99%

Isomorphisms and anti-isomorphisms of endomorphism rings of modules

Mikhalev¹

1996

First International Tainan-Moscow Algebra Workshop

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“…Such studies for automorphism groups of algebraic systems, endomorphism semigroups of graphs, endomorphism rings of modules, and other derivative algebraic systems were very successfully carried out by B. I. Plotkin [1], A. G. Pinus [2,3], L. M. Gluskin [4,5], Yu. M. Vazhenin [6,7], A. V. Mikhalev [8] and many other algebraists. S. Ulam in his well-known book [9] has mentioned the problem of characterization of mathematical objects by their endomorphisms and automorphisms.…”

Section: Introductionmentioning

confidence: 99%

On Definability of Universal Graphic Automata by Their Input Symbol Semigroups

Молчанов¹,

Farakhutdinov²

2020

MMI

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Universal graphic automaton Atm(G, G ′ ) is the universally attracting object in the category of automata, for which the set of states is equipped with the structure of a graph G and the set of output symbols is equipped with the structure of a graph G ′ preserved by transition and output functions of the automata. The input symbol semigroup of the automaton is S(G, G ′ ) = End G×Hom(G, G ′ ). It can be considered as a derivative algebraic system of the mathematical object Atm(G, G ′ ) which contains useful information about the initial automaton. It is common knowledge that properties of the semigroup are interconnected with properties of the algebraic structure of the automaton. Hence, we can study universal graphic automata by researching their input symbol semigroups. For these semigroups it is interesting to study the problem of definability of universal graphic automata by their input symbol semigroupsunder which conditions are the input symbol semigroups of universal graphic automata isomorphic. This is the subject we investigate in the present paper. The main result of our study states that the input symbol semigroups of universal graphic automata over reflexive graphs determine the initial automata up to isomorphism and duality of graphs if the state graphs of the automata contain an edge that does not belong to any cycle.

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B

Hazewinkel¹

1994

Encyclopaedia of Mathematics

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Endomorphism rings of modules and lattices of submodules

Cited by 10 publications

References 387 publications

Isomorphisms and anti-isomorphisms of endomorphism rings of modules

Isomorphisms and anti-isomorphisms of endomorphism rings of modules

On Definability of Universal Graphic Automata by Their Input Symbol Semigroups

B

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