Simple semigroup graded rings

Nystedt, Patrik; Öinert, Johan

doi:10.1142/s0219498815501029

Cited by 8 publications

(5 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is convenient to remark that the next result was originally proved when G was abelian; the authors pointed out in [22,Remark 18] that it could be proved for hypercentral groups and twisted partial actions on rings with local units. Using Lemma 3.20, the result can be proved for twisted partial actions on a unital ring R such that the ideals D g are right s-unital for all g ∈ G \ {e}.…”

Section: Proof Suppose That There Existsmentioning

confidence: 99%

Simplicity of Crossed Products by Twisted Partial Actions

Baraviera¹,

Cortes²,

Soares³

2019

J. Aust. Math. Soc.

View full text Add to dashboard Cite

In this article, we consider a twisted partial action $\unicode[STIX]{x1D6FC}$ of a group $G$ on an associative ring $R$ and its associated partial crossed product $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$. We provide necessary and sufficient conditions for the commutativity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$ when the twisted partial action $\unicode[STIX]{x1D6FC}$ is unital. Moreover, we study necessary and sufficient conditions for the simplicity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$ in the following cases: (i) $G$ is abelian; (ii) $R$ is maximal commutative in $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$; (iii) $C_{R\ast _{\unicode[STIX]{x1D6FC}}^{w}G}(Z(R))$ is simple; (iv) $G$ is hypercentral. When $R=C_{0}(X)$ is the algebra of continuous functions defined on a locally compact and Hausdorff space $X$, with complex values that vanish at infinity, and $C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$ is the associated partial skew group ring of a partial action $\unicode[STIX]{x1D6FC}$ of a topological group $G$ on $C_{0}(X)$, we study the simplicity of $C_{0}(X)\ast _{\unicode[STIX]{x1D6FC}}G$ by using topological properties of $X$ and the results about the simplicity of $R\ast _{\unicode[STIX]{x1D6FC}}^{w}G$.

show abstract

Section: Proof Suppose That There Existsmentioning

confidence: 99%

Simplicity of Crossed Products by Twisted Partial Actions

Baraviera¹,

Cortes²,

Soares³

2019

J. Aust. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Suppose that S is a unital Z-graded ring. Next, we state a special case of [17,Theorem 1.2] and [13,Theorem 5]. For the convenience of the reader, we include a shortened version of the proof from these sources adapted to the situation at hand.…”

Section: Simple Z-graded Ringsmentioning

confidence: 99%

Simplicity of Leavitt Path Algebras via Graded Ring Theory

Nystedt¹,

Öinert

2023

Bull. Aust. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Suppose that R is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Using results from graded ring theory, we show that the associated Leavitt path algebra $L_R(E)$ is simple if and only if R is simple, $E^0$ has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra.

show abstract

“…More recent developments on the ring theoretic properties include the study of the simplicity of the partial crossed products in [44,183,184,245] and [246], and the chacarterization of the Leavitt path algebras as partial group rings [186], which permitted one to obtain alternative proofs for the simplicity criterion and for the Cuntz-Krieger uniqueness theorem for Leavitt path algebras (see [184,186]). Furthermore, in [35] the relations between the module properties over A, A α G and A α were studied, where A α denotes the subring of the α-invariants, whereas in [107] the related structure of the partial skew power series ring was recently considered with respect to the distributive and Bezout duo properties.…”

Section: (G U(a))→pic(a G )→Pic(a) G →H 2 (G U(a))→b(a/a α )→ → H 1mentioning

confidence: 99%

Recent developments around partial actions

Dokuchaev

2018

São Paulo J. Math. Sci.

View full text Add to dashboard Cite

We give an overview of publications on partial actions and related concepts, paying main attention to some recent developments on diverse aspects of the theory, such as partial actions of semigroups, of Hopf algebras and groupoids, the globalization problem for partial actions, Morita theory of partial actions, twisted partial actions, partial projective representations and the Schur multiplier, cohomology theories related to partial actions, Galois theoretic results, ring theoretic properties and ideals of partial crossed products. Among the applications we consider in more detail the case of the Carlsen-Matsumoto C *-algebra related to an arbitrary subshift, but also mention many others. The total number of publications directly related to partial actions and partial representations is more than 130, so that it is impossible even to describe briefly the content of all of them within the constraints of the present survey. Thus, the majority of them are only cited with respects to specific topics, trying to give an idea about the involved matter. In order to complete the picture, we refer the reader to a recent book by Ruy Exel, to our previous surveys, as well as to those by other authors.

show abstract

Simple semigroup graded rings

Cited by 8 publications

References 36 publications

Simplicity of Crossed Products by Twisted Partial Actions

Simplicity of Crossed Products by Twisted Partial Actions

Simplicity of Leavitt Path Algebras via Graded Ring Theory

Recent developments around partial actions

Contact Info

Product

Resources

About