Uniserial Noetherian centrally essential rings

Марков, В. И.; Tuganbaev, A. A.

doi:10.1080/00927872.2019.1635607

Cited by 16 publications

(8 citation statements)

References 14 publications

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“…This subsection is based on [47] and [49]. For convenience, we give brief proofs of the following two well known assertions.…”

Section: Uniserial Noetherian Ringsmentioning

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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“…This subsection is based on [47] and [49]. For convenience, we give brief proofs of the following two well known assertions.…”

Section: Uniserial Noetherian Ringsmentioning

confidence: 99%

Centrally Essential Rings and Semirings

Tuganbaev¹

2022

Preprint

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“…Any non-zero ring with zero multiplication is centrally essential but does not satisfy ( * ). 1 Centrally essential rings with non-zero identity element are studied in papers [6], [7], [8], [9], [10], [11], [12]. Every centrally essential semiprime ring with 1 = 0 is commutative; see [6,Proposition 3.3].…”

Section: Centrally Essential Ringsmentioning

confidence: 99%

Local Centrally Essential Subalgebras of Triangular Algebras

Lyubimtsev

Tuganbaev

2020

Preprint

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We study local centrally essential subalgebras in the algebra of all upper triangular matrices over a field of characteristic = 2. It is proved that the algebras of upper triangular 3 × 3 or 4 × 4 matrices have only commutative local centrally essential subalgebras. Every algebra of upper triangular matrices of order exceeding 6 contains a non-commutative local centrally essential subalgebra. 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. MSC2010 database 16R99; 20K30 1 Not necessarily unital rings with ( * ) are considered in [13].

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“…A ring R is said to be centrally essential if for every non-zero element a ∈ R, there exist non-zero central elements x, y with ax = y. 3 Centrally essential rings with non-zero identity elements are studied in [6], [7], [8], [9], [10], [11], [12]. Every centrally essential semiprime ring with 1 = 0 is commutative; see [6,Proposition 3.3].…”

Section: Centrally Essential Ringsmentioning

confidence: 99%

Centrally Essential Torsion-Free Rings of Finite Rank

Lyubimtsev¹,

Tuganbaev²

2020

Preprint

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It is proved that centrally essential rings, whose additive groups of finite rank are torsion-free groups of finite rank, are quasi-invariant but not necessarily invariant. Torsion-free Abelian groups of finite rank with centrally essential endomorphism rings are faithful.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.

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Uniserial Noetherian centrally essential rings

Cited by 16 publications

References 14 publications

Centrally Essential Rings and Semirings

Centrally Essential Rings and Semirings

Local Centrally Essential Subalgebras of Triangular Algebras

Centrally Essential Torsion-Free Rings of Finite Rank

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