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 Étale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings

2020 ◽  
pp. 1-35
Author(s):  
Daniel Gonçalves ◽  
Benjamin Steinberg
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Ring ◽  
Semigroup Rings ◽  
Convolution Algebra ◽  
Special Cases ◽  
Unital Rings ◽  
Étale Groupoid

Abstract Given an action ${\varphi }$ of inverse semigroup S on a ring A (with domain of ${\varphi }(s)$ denoted by $D_{s^*}$ ), we show that if the ideals $D_e$ , with e an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.

scholarly journals Formal complexity of inverse semigroup rings

1990 ◽  
Vol 41 (3) ◽  
pp. 343-346 ◽  
Author(s):  
Adel A. Shehadah
Keyword(s):  
Inverse Semigroup ◽  
Semigroup Ring ◽  
Sufficient Condition ◽  
Semigroup Rings

A ring (R, *) with involution * is called formally complex if implies that all Ai are 0. Let (R, *) be a formally complex ring and let S be an inverse semigroup. Let (R[S], *) be the semigroup ring with involution * defined by . We show that (R[S], *) is a formally complex ring. Let (S, *) be a semigroup with proper involution *(aa* = ab* = bb* ⇒ a = b) and let (R, *′) be a formally complex ring. We give a sufficient condition for (R[S], *′) to be a formally complex ring and this condition is weaker than * being the inverse involution on S. We illustrate this by an example.

scholarly journals The strong nil-cleanness of semigroup rings

2020 ◽  
Vol 18 (1) ◽  
pp. 1491-1500
Author(s):  
Yingdan Ji
Keyword(s):  
Inverse Semigroup ◽  
Simple Semigroup ◽  
Semigroup Ring ◽  
Semigroup Rings ◽  
Locally Inverse Semigroup ◽  
Semilattice Of Semigroups ◽  
Strong Semilattice ◽  
Alpha Beta ◽  
Strong Semilattice Of Semigroups

Abstract In this paper, we study the strong nil-cleanness of certain classes of semigroup rings. For a completely 0-simple semigroup M={ {\mathcal M} }^{0}(G;I,\text{Λ};P) , we show that the contracted semigroup ring {R}_{0}{[}M] is strongly nil-clean if and only if either |I|=1 or |\text{Λ}|=1 , and R{[}G] is strongly nil-clean; as a corollary, we characterize the strong nil-cleanness of locally inverse semigroup rings. Moreover, let S={[}Y;{S}_{\alpha },{\varphi }_{\alpha ,\beta }] be a strong semilattice of semigroups, then we prove that R{[}S] is strongly nil-clean if and only if R{[}{S}_{\alpha }] is strongly nil-clean for each \alpha \in Y .

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 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 Chain conditions and semigroup graded rings

1988 ◽  
Vol 45 (3) ◽  
pp. 372-380 ◽  
Author(s):  
E. Jespers
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Cancellative Semigroup ◽  
Graded Rings ◽  
Semigroup Rings ◽  
Graded Ring ◽  
Product Group ◽  
Chain Conditions ◽  
Unique Product

AbstractThe following questions are studied: When is a semigroup graded ring left Noetherian, respectively semiprime left Goldie? Necessary sufficient conditions are proved for cancellative semigroup-graded subrings of rings weakly or strongly graded by a polycyclic-by-finite (unique product) group. For semigroup rings R[S] we also give a solution to the problem in case S is an inverse semigroup.

scholarly journals Continuous Orbit Equivalence on Self-Similar Graph Actions

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 990
Author(s):  
Inhyeop Yi
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Orbit Equivalence ◽  
Semigroup Actions ◽  
Infinite Path ◽  
Self Similar

For self-similar graph actions, we show that isomorphic inverse semigroups associated to a self-similar graph action are a complete invariant for the continuous orbit equivalence of inverse semigroup actions on infinite path spaces.

scholarly journals Brandt Extensions and Primitive Topological Inverse Semigroups

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Tetyana Berezovski ◽  
Oleg Gutik ◽  
Kateryna Pavlyk
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Inverse Semigroup ◽  
Countably Compact ◽  
Absolutely Closed

We study (countably) compact and (absolutely) -closed primitive topological inverse semigroups. We describe the structure of compact and countably compact primitive topological inverse semigroups and show that any countably compact primitive topological inverse semigroup embeds into a compact primitive topological inverse semigroup.

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.

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.

Sign in / Sign up

Export Citation Format

Share Document