Hausdorff series in a semigroup ring

2020 ◽  
Vol 30 (04) ◽  
pp. 853-859
Author(s):  
Şehmus Fındık ◽  
Osman Kelekci̇
Keyword(s):  
Identity Element ◽  
Semigroup Ring ◽  
Left Zero ◽  
Zero Semigroup ◽  
Closed Formula ◽  
Semigroup Rings ◽  
Left Zero Semigroup ◽  
The Right

Let [Formula: see text] and [Formula: see text] be the semigroup rings spanned on the right zero semigroup [Formula: see text], and on the left zero semigroup [Formula: see text], respectively, together with the identity element [Formula: see text]. We suggest a closed formula solving the equation [Formula: see text] which is the evolution of the Campbell–Baker–Hausdorff formula given by the Hausdorff series [Formula: see text] where [Formula: see text], in the algebras [Formula: see text] and [Formula: see text].

Independent domination number in Cayley digraphs of rectangular groups

2018 ◽  
Vol 10 (02) ◽  
pp. 1850024
Author(s):  
Nuttawoot Nupo ◽  
Sayan Panma
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Left Zero ◽  
Zero Semigroup ◽  
Left Zero Semigroup ◽  
Group A ◽  
Independent Domination ◽  
Independent Dominating Set ◽  
Rectangular Group

Let [Formula: see text] denote the Cayley digraph of the rectangular group [Formula: see text] with respect to the connection set [Formula: see text] in which the rectangular group [Formula: see text] is isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. An independent dominating set of [Formula: see text] is the independent set of elements in [Formula: see text] that can dominate the whole elements. In this paper, we investigate the independent domination number of [Formula: see text] and give more results on Cayley digraphs of left groups and right groups which are specific cases of rectangular groups. Moreover, some results of the path independent domination number of [Formula: see text] are also shown.

scholarly journals CAYLEY AUTOMATON SEMIGROUPS

2009 ◽  
Vol 19 (01) ◽  
pp. 79-95 ◽  
Author(s):  
VICTOR MALTCEV
Keyword(s):  
Left Zero ◽  
Zero Semigroup ◽  
Left Zero Semigroup ◽  
Automaton Semigroups

In this paper we characterize when a Cayley automaton semigroup is finite, is free, is a left zero semigroup, is a right zero semigroup, is a group, or is trivial. We also introduce dual Cayley automaton semigroups and discuss when they are finite.

On Ore extension and skew power series rings with some restrictions on zero-divisors

2016 ◽  
Vol 16 (09) ◽  
pp. 1750164
Author(s):  
E. Hashemi ◽  
A. As. Estaji ◽  
A. Alhevaz
Keyword(s):  
Power Series ◽  
Left Zero ◽  
Ore Extension ◽  
Series Ring ◽  
Zero Divisors ◽  
Open Questions ◽  
Ore Extensions ◽  
Power Series Rings ◽  
Extension Ring ◽  
The Right

The study of rings with right Property ([Formula: see text]), has done an important role in noncommutative ring theory. Following literature, a ring [Formula: see text] has right Property ([Formula: see text]) if every finitely generated two-sided ideal consisting entirely of left zero-divisors has a nonzero right annihilator. Our results in this paper concerns the right Property ([Formula: see text]) of Ore extensions as well as skew power series rings. We will show that if [Formula: see text] is a right duo ring, then the skew power series ring [Formula: see text] has right Property ([Formula: see text]), when [Formula: see text] is right Noetherian and [Formula: see text]-compatible. Moreover, for a right duo ring [Formula: see text] which is [Formula: see text]-compatible, it is shown that (i) the Ore extension ring [Formula: see text] has right Property ([Formula: see text]) and (ii) [Formula: see text] is right zip if and only if [Formula: see text] is right zip. As a corollary of our results, we provide answers to some open questions related to Property [Formula: see text], raised in [C. Y. Hong, N. K. Kim, Y. Lee and S. J. Ryu, Rings with Property ([Formula: see text]) and their extensions, J. Algebra 315 (2007) 612–628].

Arithmetical Semigroup Rings

1980 ◽  
Vol 32 (6) ◽  
pp. 1361-1371 ◽  
Author(s):  
Bonnie R. Hardy ◽  
Thomas S. Shores
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Forthcoming Paper ◽  
Semigroup Ring ◽  
Necessary And Sufficient Conditions ◽  
Semigroup Rings ◽  
Principal Ideal Ring ◽  
Arithmetical Semigroup ◽  
Necessary And Sufficient

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and S that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4].

scholarly journals Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics

2019 ◽  
Vol 31 (3) ◽  
pp. 543-562 ◽  
Author(s):  
Viviane Beuter ◽  
Daniel Gonçalves ◽  
Johan Öinert ◽  
Danilo Royer
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Ring ◽  
Topological Dynamics ◽  
Topological Properties ◽  
Semigroup Rings ◽  
Partial Action ◽  
Semigroup Actions ◽  
Formula One ◽  
Topological Inverse Semigroup ◽  
Commutative Subring

Abstract Given a partial action π of an inverse semigroup S on a ring {\mathcal{A}} , one may construct its associated skew inverse semigroup ring {\mathcal{A}\rtimes_{\pi}S} . Our main result asserts that, when {\mathcal{A}} is commutative, the ring {\mathcal{A}\rtimes_{\pi}S} is simple if, and only if, {\mathcal{A}} is a maximal commutative subring of {\mathcal{A}\rtimes_{\pi}S} and {\mathcal{A}} is S-simple. We apply this result in the context of topological inverse semigroup actions to connect simplicity of the associated skew inverse semigroup ring with topological properties of the action. Furthermore, we use our result to present a new proof of the simplicity criterion for a Steinberg algebra {A_{R}(\mathcal{G})} associated with a Hausdorff and ample groupoid {\mathcal{G}} .

scholarly journals On Segre products of affine semigroup rings

1988 ◽  
Vol 110 ◽  
pp. 113-128 ◽  
Author(s):  
Lê Tuân Hoa
Keyword(s):  
Positive Integer ◽  
Polynomial Ring ◽  
Semigroup Ring ◽  
Semigroup Rings ◽  
Finitely Generated ◽  
Affine Semigroup

Let N denote the set of non-negative integers. An affine semigroup is a finitely generated submonoid S of the additive monoid Nm for some positive integer m. Let k[S] denote the semigroup ring of S over a field k. Then one can identify k[S] with the subring of a polynomial ring k[t1, …, tm] generated by the monomials .

scholarly journals On completely0-simple semigroups

1996 ◽  
Vol 19 (3) ◽  
pp. 507-520 ◽  
Author(s):  
Yue-Chan Phoebe Ho
Keyword(s):  
Simple Semigroup ◽  
Matrix Representation ◽  
Semigroup Ring ◽  
Closed Field ◽  
Zero Semigroup ◽  
Algebraically Closed Field

LetSbe a completely0-simple semigroup andFbe an algebraically closed field. Then for each0-minimal right idealMofS,M=B∪C∪{0}, whereBis a right group andCis a zero semigroup. Also, a matrix representation forSother than Rees matrix is found for the condition that the semigroup ringR(F,S)is semisimple Artinian.

NORMALIZERS AND FREE GROUPS IN THE UNIT GROUP OF AN INTEGRAL SEMIGROUP RING

2007 ◽  
Vol 06 (04) ◽  
pp. 655-669 ◽  
Author(s):  
ANN DOOMS ◽  
PAULA M. VELOSO
Keyword(s):  
Free Groups ◽  
Unit Group ◽  
Semigroup Ring ◽  
Inverse Semigroups ◽  
Brandt Semigroup ◽  
Group Rings ◽  
Semigroup Rings ◽  
Natural Basis ◽  
Partial Group ◽  
Matrix Rings

In this article, we introduce the normalizer [Formula: see text] of a subset X of a ring R (with identity) in the unit group [Formula: see text] and consider, in particular, the normalizer of the natural basis ±S of the integral semigroup ring ℤ0S of a finite semigroup S. We investigate properties of this normalizer for the class of semigroup rings of inverse semigroups, which contains, for example, matrix rings, in particular, matrix rings over group rings, and partial group rings. We also construct free groups in the unit group of an integral semigroup ring of a Brandt semigroup using a bicyclic unit.

Cycloidal algebras

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Jeong Han ◽  
Hee Kim ◽  
J. Neggers
Keyword(s):  
Binary System ◽  
Golden Section ◽  
Commutative Rings ◽  
Zero Semigroup ◽  
Real Numbers ◽  
The Real ◽  
Finite Algebras ◽  
The Right ◽  
Linear Product

AbstractIn this paper we introduce for an arbitrary algebra (groupoid, binary system) (X; *) a sequence of algebras (X; *)n = (X; ∘), where x ∘ y = [x * y]n = x * [x * y]n−1, [x * y]0 = y. For several classes of examples we study the cycloidal index (m, n) of (X; *), where (X; *)m = (X; *)n for m > n and m is minimal with this property. We show that (X; *) satisfies the left cancellation law, then if (X; *)m = (X; *)n, then also (X; *)m−n = (X; *)0, the right zero semigroup. Finite algebras are shown to have cycloidal indices (as expected). B-algebras are considered in greater detail. For commutative rings R with identity, x * y = ax + by + c, a, b, c ∈ ℝ defines a linear product and for such linear products the commutativity condition [x * y]n = [y * x]n is observed to be related to the golden section, the classical one obtained for ℝ, the real numbers, n = 2 and a = 1 as the coefficient b.

scholarly journals Semiprime semigroup rings and a problem of J. Weissglass

1980 ◽  
Vol 21 (1) ◽  
pp. 131-134 ◽  
Author(s):  
Mark L. Teply
Keyword(s):  
Semigroup Ring ◽  
Semigroup Rings

If R is a ring and S is a semigroup, the corresponding semigroup ring is denoted by R[S]. A ring is semiprime if it has no nonzero nilpotent ideals. A semigroup S is a semilattice P of semigroups Sα if there exists a homomorphism φ of S onto the semilattice P such that Sα = αφ−1 for each α ∈ P.

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