E-Groups and E-Rings
An associative ring R is called an E-ring if the canonical homomorphism R ≅ E( R + ) is an isomorphism. Additive groups of E -rings are called E-groups . In other words, an Abelian group A is an E -group if and only if A ≅ End A and the endomorphism ring E( A ) is commutative. In this paper, we give...
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| Published in | Journal of mathematical sciences (New York, N.Y.) Vol. 256; no. 3; pp. 341 - 361 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
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Springer US
04.07.2021
Springer Springer Nature B.V |
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| Abstract | An associative ring
R
is called an
E-ring
if the canonical homomorphism
R
≅ E(
R
+
) is an isomorphism. Additive groups of
E
-rings are called
E-groups
. In other words, an Abelian group
A
is an
E
-group if and only if
A
≅ End
A
and the endomorphism ring E(
A
) is commutative. In this paper, we give a survey of the main results on
E
-groups and
E
-rings and also consider some of their generalizations:
ε
-closed groups,
T
-rings,
A
-rings, the groups admitting only commutative multiplications, etc. |
|---|---|
| AbstractList | An associative ring R is called an E-ring if the canonical homomorphism R [congruent to] E([R.sup.+]) is an isomorphism. Additive groups of E-rings are called E-groups. In other words, an Abelian group A is an E-group if and only if A [congruent to] End A and the endomorphism ring E(A) is commutative. In this paper, we give a survey of the main results on E-groups and E-rings and also consider some of their generalizations: [epsilon]-closed groups, T-rings, A-rings, the groups admitting only commutative multiplications, etc. An associative ring R is called an E-ring if the canonical homomorphism R ≅ E( R + ) is an isomorphism. Additive groups of E -rings are called E-groups . In other words, an Abelian group A is an E -group if and only if A ≅ End A and the endomorphism ring E( A ) is commutative. In this paper, we give a survey of the main results on E -groups and E -rings and also consider some of their generalizations: ε -closed groups, T -rings, A -rings, the groups admitting only commutative multiplications, etc. An associative ring R is called an E-ring if the canonical homomorphism R [congruent to] E([R.sup.+]) is an isomorphism. Additive groups of E-rings are called E-groups. In other words, an Abelian group A is an E-group if and only if A [congruent to] End A and the endomorphism ring E(A) is commutative. In this paper, we give a survey of the main results on E-groups and E-rings and also consider some of their generalizations: [epsilon]-closed groups, T-rings, A-rings, the groups admitting only commutative multiplications, etc. Keywords and phrases: Abelian group, [epsilon]-closed group, E-group, E-ring, T-ring, quotient divisible group, A-ring, endomorphism ring. AMS Subject Classification: 20Kxx An associative ring R is called an E-ring if the canonical homomorphism R ≅ E(R+) is an isomorphism. Additive groups of E-rings are called E-groups. In other words, an Abelian group A is an E-group if and only if A ≅ End A and the endomorphism ring E(A) is commutative. In this paper, we give a survey of the main results on E-groups and E-rings and also consider some of their generalizations: ε-closed groups, T-rings, A-rings, the groups admitting only commutative multiplications, etc. |
| Audience | Academic |
| Author | Tuganbaev, A. A. Krylov, P. A. Tsarev, A. V. |
| Author_xml | – sequence: 1 givenname: P. A. surname: Krylov fullname: Krylov, P. A. email: krylov@math.tsu.ru organization: National Research Tomsk State University – sequence: 2 givenname: A. A. surname: Tuganbaev fullname: Tuganbaev, A. A. organization: Moscow Power Engineering Institute (National Research University), M. V. Lomonosov Moscow State University – sequence: 3 givenname: A. V. surname: Tsarev fullname: Tsarev, A. V. organization: Moscow Pedagogical State University |
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Zh. – ident: 5430_CR19 – ident: 5430_CR34 – ident: 5430_CR15 – volume: 21 start-page: 132 year: 1970 ident: 5430_CR30 publication-title: Arch. Math. doi: 10.1007/BF01220893 |
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| Snippet | An associative ring
R
is called an
E-ring
if the canonical homomorphism
R
≅ E(
R
+
) is an isomorphism. Additive groups of
E
-rings are called
E-groups
. In... An associative ring R is called an E-ring if the canonical homomorphism R [congruent to] E([R.sup.+]) is an isomorphism. Additive groups of E-rings are called... An associative ring R is called an E-ring if the canonical homomorphism R ≅ E(R+) is an isomorphism. Additive groups of E-rings are called E-groups. In other... |
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| StartPage | 341 |
| SubjectTerms | Associativity Electronic discussion groups Group theory Homomorphisms Isomorphism Mathematics Mathematics and Statistics Rings (mathematics) Surveys |
| Title | E-Groups and E-Rings |
| URI | https://link.springer.com/article/10.1007/s10958-021-05430-2 https://www.proquest.com/docview/2544895492 |
| Volume | 256 |
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