On the radical of semigroup algebras satisfying polynomial identities

1986 ◽  
Vol 99 (1) ◽  
pp. 45-50 ◽  
Author(s):  
Jan Okniński
Keyword(s):  
Commutative Semigroup ◽  
Polynomial Identity ◽  
Jacobson Radical ◽  
Semigroup Algebra ◽  
Group Algebras ◽  
Semigroup Algebras ◽  
Polynomial Identities ◽  
Arbitrary Semigroup

In this paper we will be concerned with the problem of describing the Jacobson radical of the semigroup algebraK[S] of an arbitrary semigroupSover a fieldKin the case where this algebra satisfies a polynomial identity. Recently, Munn characterized the radical of commutative semigroup algebras [9]. The key to his result was to show that, in this situation, the radical must be a nilideal. We are going to extend the latter to the case of PI-semigroup algebras. Further, we characterize the radical by means of the properties ofSor, more precisely, by some groups derived fromS. For this purpose we will exploit earlier results leading towards a characterization of semigroup algebras satisfying polynomial identities [5], [15], which generalized the well known case of group algebras (cf. [13], chap. 5).

scholarly journals On group graded rings satisfying polynomial identities

1995 ◽  
Vol 37 (2) ◽  
pp. 205-210 ◽  
Author(s):  
A. V. Kelarev ◽  
J. Okniński
Keyword(s):  
Commutative Semigroup ◽  
Jacobson Radical ◽  
Normal Band ◽  
Ring Theory ◽  
Semigroup Ring ◽  
Semigroup Algebras ◽  
Graded Rings ◽  
Finitely Generated ◽  
Polynomial Identities ◽  
Locally Finite

A number of classical theorems of ring theory deal with nilness and nilpotency of the Jacobson radical of various ring constructions (see [10], [18]). Several interesting results of this sort have appeared in the literature recently. In particular, it was proved in [1] that the Jacobson radical of every finitely generated PI-ring is nilpotent. For every commutative semigroup ring RS, it was shown in [11] that if J(R) is nil then J(RS) is nil. This result was generalized to all semigroup algebras satisfying polynomial identities in [15] (see [16, Chapter 21]). Further, it was proved in [12] that, for every normal band B, if J(R) is nilpotent, then J(RB) is nilpotent. A similar result for special band-graded rings was established in [13, Section 6]. Analogous theorems concerning nilpotency and local nilpotency were proved in [2] for rings graded by finite and locally finite semigroups.

scholarly journals Group algebras and semigroup algebras defined by permutation relations of fixed length

2015 ◽  
Vol 15 (02) ◽  
pp. 1650037
Author(s):  
Ferran Cedó ◽  
Eric Jespers ◽  
Georg Klein
Keyword(s):  
Symmetric Group ◽  
Finite Index ◽  
Jacobson Radical ◽  
Sufficient Conditions ◽  
Semigroup Algebra ◽  
Free Subgroup ◽  
Group Algebras ◽  
Semigroup Algebras ◽  
Fixed Integer ◽  
Kirillov Dimension

Let H be a subgroup of Sym n, the symmetric group of degree n. For a fixed integer l ≥ 2, the group G presented with generators x1, x2,…,xn and with relations xi1xi2⋯xil = xσ(i1) xσ(i2)⋯xσ(il), where σ runs through H, is considered. It is shown that G has a free subgroup of finite index. For a field K, properties of the algebra K[G] are derived. In particular, the Jacobson radical 𝒥(K[G]) is always nilpotent, and in many cases the algebra K[G] is semiprimitive. Results on the growth and the Gelfand–Kirillov dimension of K[G] are given. Further properties of the semigroup S and the semigroup algebra K[S] with the same presentation are obtained, in case S is cancellative. The Jacobson radical is nilpotent in this case as well, and sufficient conditions for the algebra to be semiprimitive are given.

On the Jacobson radical of certain commutative semigroup algebras

1984 ◽  
Vol 96 (1) ◽  
pp. 15-23 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Commutative Ring ◽  
Commutative Semigroup ◽  
Jacobson Radical ◽  
Semigroup Algebras

In two previous papers the author studied the Jacobson and nil redicals of the algebra of a commutative semigroup over a field [8] and over a commutative ring with unity [9]. This work is continued here.

scholarly journals Module amenability of restricted semigroup algebras under module actions

Filomat ◽  
2015 ◽  
Vol 29 (4) ◽  
pp. 787-793
Author(s):  
Abbas Sahleh ◽  
Somaye Tanha
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Clifford Semigroup ◽  
Module Amenability

In this article, weshow that module amenability with the canonical action of restricted semigroup algebra l1r (S) and semigroup algebra l1(Sr) are equivalent, where Sr is the restricted semigroup of associated to the inverse semigroup S. We use this to give a characterization of module amenability of restricted semigroup algebra l1r (S) with the canonical action, where S is a Clifford semigroup.

Compact multipliers on weighted hypergroup algebras

1985 ◽  
Vol 98 (3) ◽  
pp. 493-500 ◽  
Author(s):  
F. Ghahramani ◽  
A. R. Medgalchi
Keyword(s):  
Banach Algebra ◽  
Group Algebras ◽  
Semigroup Algebras ◽  
Weighted Semigroup ◽  
Hypergroup Algebras

AbstractLet Mω(X) be a weighted hypergroup algebra, and Lω(X) be the Banach algebra of measures μ ε Mω(X) such that the function x ↦ (1/ω(x))δx* |μ| is norm continuous. We characterize compact multipliers on Lω(X). This extends the characterization of compact multipliers on weighted group algebras and some classes of weighted semigroup algebras.

Locally ample semigroup algebras

2014 ◽  
Vol 07 (04) ◽  
pp. 1450067 ◽  
Author(s):  
Xiaojiang Guo ◽  
K. P. Shum
Keyword(s):  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Abundant Semigroup ◽  
Ample Semigroup

An idempotent-connected abundant semigroup S is a locally ample semigroup if for any idempotent e of S, the local submonoid eSe of S is an ample subsemigroup of S. Clearly, an ample semigroup is a locally ample semigroup. In this paper, it is proved that the semigroup algebra of a finite locally ample semigroup is isomorphic to the semigroup algebra of an associate primitive abundant semigroup. As an application of this result, we characterize Jacobson radicals of finite locally ample semigroup algebras.

THE FIRST COHOMOLOGY GROUP OF BANACH INVERSE SEMIGROUP ALGEBRAS WITH COEFFICIENTS IN -EMBEDDED BANACH BIMODULES

2019 ◽  
Vol 101 (3) ◽  
pp. 488-495
Author(s):  
HOGER GHAHRAMANI
Keyword(s):  
Banach Space ◽  
Inverse Semigroup ◽  
Cohomology Group ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Mild Conditions ◽  
First Cohomology Group

Let $S$ be a discrete inverse semigroup, $l^{1}(S)$ the Banach semigroup algebra on $S$ and $\mathbb{X}$ a Banach $l^{1}(S)$-bimodule which is an $L$-embedded Banach space. We show that under some mild conditions ${\mathcal{H}}^{1}(l^{1}(S),\mathbb{X})=0$. We also provide an application of the main result.

scholarly journals Self-Dual Abelian Codes in Some Nonprincipal Ideal Group Algebras

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Parinyawat Choosuwan ◽  
Somphong Jitman ◽  
Patanee Udomkavanich
Keyword(s):  
Cyclic Codes ◽  
Group Algebras ◽  
Dual Codes ◽  
Complete Enumeration ◽  
Odd Order ◽  
Abelian Codes ◽  
Positive Integers ◽  
Ideal Group ◽  
Suitable Product

The main focus of this paper is the complete enumeration of self-dual abelian codes in nonprincipal ideal group algebrasF2k[A×Z2×Z2s]with respect to both the Euclidean and Hermitian inner products, wherekandsare positive integers andAis an abelian group of odd order. Based on the well-known characterization of Euclidean and Hermitian self-dual abelian codes, we show that such enumeration can be obtained in terms of a suitable product of the number of cyclic codes, the number of Euclidean self-dual cyclic codes, and the number of Hermitian self-dual cyclic codes of length2sover some Galois extensions of the ringF2k+uF2k, whereu2=0. Subsequently, general results on the characterization and enumeration of cyclic codes and self-dual codes of lengthpsoverFpk+uFpkare given. Combining these results, the complete enumeration of self-dual abelian codes inF2k[A×Z2×Z2s]is therefore obtained.

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