Centrally Essential Group Algebras and Classical Rings of Fractions

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

doi:10.1134/s1995080221120246

Cited by 4 publications

(6 citation statements)

References 12 publications

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“…The following theorem extends the result of [4,Theorem 1] to semigroup algebras of cancellative semigroups. In what follows F is a field, S is a semigroup, F S is the semigroup algebra of the semigroup S over the field F .…”

Section: Introductionsupporting

confidence: 54%

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Centrally Essential Semigroup Algebras

Lyubimtsev¹,

Tuganbaev²

2022

Preprint

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For a cancellative semigroup S and a field F, it is proved that the semigroup algebra FS is centrally essential if and only if the group of fractions G S of the semigroup S exists and the group algebra F G S of G S is centrally essential. The semigroup algebra of a cancellative semigroup is centrally essential if and only if it has the classical right ring of fractions which is a centrally essential ring. There exist non-commutative centrally essential semigroup algebras over fields of zero characteristic (this contrasts with the known fact that centrally essential group algebras over fields of zero characteristic are commutative).

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Section: Introductionsupporting

confidence: 54%

“…The semigroup S has the group of fractions G S which is a free nilpotent group of nilpotence class 2; see [8,Example 24.21]. It is known that if a group does not contain elements of order p, then centrally essential group algebra is commutative; see [4,Proposition 1]. Therefore, the group algebra F G S is not centrally essential.…”

Section: 1mentioning

confidence: 99%

Centrally Essential Semigroup Algebras

Lyubimtsev¹,

Tuganbaev²

2022

Preprint

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“…The semigroup S has the group of fractions G S which is a free nilpotent group of nilpotence class 2; see [54,Example 21]. It follows known that if the group does not contain of elements of order p, then centrally essential group algebra is commutative; see [36,Proposition 1]. Therefore, the group algebra F G S is not centrally essential.…”

Section: Rings Of Fractions and Semigroup 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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“…x = α 0 • 1 + α 1 1 e 1 + α 2 1 e 2 + α 3 1 e 3 + α 1 2 e 1 ∧ e 2 + α 2 2 e 1 ∧ e 3 + +α 3 2 e 2 ∧ e 3 + α 3 e 1 ∧ e 2 ∧ e 3 , then [e 1 , x] = 2α 2 1 e 1 ∧ e 2 + 2α 3 1 e 1 ∧ e 3 , [e 2 , x] = − 2α 1 1 e 1 ∧ e 2 + 2α 3 1 e 2 ∧ e 3 , [e 3 , x] = − 2α 1 1 e 1 ∧ e 3 − 2α 2 1 e 2 ∧ e 3 . Thus, x ∈ Z(Λ(V)) if and only if α 1 1 = α 2 1 = α 3 1 = 0.…”

Section: Therefore Ifunclassified

“…The work is based on papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. The author is very grateful for the great help in editing the manuscript to Adel Abyzov, Oleg Lyubimtsev and Danil Tapkin.…”

Section: Introductionmentioning

confidence: 99%