Local centrally essential subalgebras of triangular algebras

Lyubimtsev, O. V.; Tuganbaev, A. A.

doi:10.1080/03081087.2020.1802402

Cited by 6 publications

(8 citation statements)

References 14 publications

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“…However, the ring of differences D(S) = M 3 (Z) is not a centrally essential ring, since the ring has non-central idempotents. In addition, any centrally essential subalgebra of a local triangular 3 × 3 matrix algebra is commutative; this is proved in [7].…”

Section: Additively Cancellative Centrally Essential Semiringsmentioning

confidence: 99%

“…over the ring Z of integers. In [7], it is proved that R is a non-commutative centrally essential ring. Let S 1 be the semiring generated by matrices of the form (1) over Z + and scalar matrices with α ∈ Z + ∪ {0} and zeros на the remaining positions.…”

Section: Examplementioning

confidence: 99%

“…In [13], a centrally essential ring R is constructed such that the factor ring R/J(R) with respect to the Jacobson radical is not a P I ring (in particular, the ring R/J(R) is not commutative). Matrix centrally essential algebras are studied in [7]. Abelian groups with centrally essential endomorphism rings are considered in [6] and [8].…”

Section: Introductionmentioning

confidence: 99%

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Centrally Essential Semirings

Lyubimtsev¹,

Tuganbaev²

2022

Preprint

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A semiring is said to be centrally essential if for every non-zero element x, there exist two non-zero central elements y, z with xy = z. We give some examples of non-commutative centrally essential semirings and describe some properties of additively cancellative centrally essential semirings.The work of O.V. Lyubimtsev is done under the state assignment No 0729-2020-0055. A.A.Tuganbaev is supported by Russian Scientific Foundation, project 16-11-10013P.1.1. Example. We consider a semigroup (M, •) with multiplication table

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Section: Additively Cancellative Centrally Essential Semiringsmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

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Centrally Essential Semirings

Lyubimtsev¹,

Tuganbaev²

2022

Preprint

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“…Therefore, the idempotent e is central. ✄ 1.1.5; [38]. Let a ring A be a not necessarily unital, centrally essential ring, e = e 2 ∈ A, a, x 1 , .…”

Section: Therefore Ifmentioning

confidence: 99%

“…This subsection is based on [38]. In this subsection, we consider not necessarily unital rings and study local centrally essential subalgebras of the algebra T n (F) of all upper of triangular matrices, where F is a field of characteristic = 2.…”

Section: Local Sublgebras Of Triangular Algebrasmentioning

confidence: 99%

Centrally Essential Rings and Semirings

Tuganbaev¹

2022

Preprint

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This work is a review of results about centrally essential rings and semirings. A ring (resp., semiring) is said to be centrally essential if it is either commutative or satisfy the property that for any non-central element a, there exist non-zero central elements x and y with ax = y. The class of centrally essential rings is very large; many corresponding examples are given in the work.non-central element a ∈ A, there are non-zero central elements x and y with ax = y.It is clear that any commutative ring is centrally essential. An unital ring A with center C = Z(A) is centrally essential if and only if the C-module A is an essential extension of C C .From the definition of a centrally essential ring A, it might seem that such a ring is possibly commutative. Indeed, A satisfies many propeties of commutative rings. For example,• all idempotents of A are central, see 1.1.4 below;• If A is a centrally essential local ring, then the ring A/J(A) is a field and, in particular, is commutative; see 1.3.2.• If A is a right or left semi-Artinian centrally essential ring, then the factor ring A/J(A) is commutative; see 1.4.5.However, a centrally essential ring A may be very far from a commutative ring. For example,

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Centrally Essential Group Algebras and Classical Rings of Fractions

Lyubimtsev

Tuganbaev

2021

Lobachevskii J Math

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Local centrally essential subalgebras of triangular algebras

Cited by 6 publications

References 14 publications

Centrally Essential Semirings

Centrally Essential Semirings

Centrally Essential Rings and Semirings

Centrally Essential Group Algebras and Classical Rings of Fractions

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