Fractional skew monoid rings

By P. Ara, M. A. González-Barroso, K. Goodearl, Enrique Pardo. 2003

62 citations recorded by the source corpus.

Abstract Given an action α of a monoid T on a ring A by ring endomorphisms, and an Ore subset S of T, a general construction of a fractional skew monoid ring S op ∗ α A∗ α T is given, extending the usual constructions of skew group rings and of skew semigroup rings. In case S is a subsemigroup of a group G such that G=S−1S, we obtain a G-graded ring S op ∗ α A∗ α S with the property that, for each s∈S, the s-component contains a left invertible element and the s−1-component contains a right invertible element. In the most basic case, where G= Z and S=T= Z + , the construction is fully determined by a single ring endomorphism α of A. If α is an isomorphism onto a proper corner pAp, we obtain an analogue of the usual skew Laurent polynomial ring, denoted by A[t+,t−;α]. Examples of this construction are given, and it is proven that several classes of known algebras, including the Leavitt algebras of type (1,n), can be presented in the form A[t+,t−;α]. Finally, mild and reasonably natural conditions are obtained under which S op ∗ α A∗ α S is a purely infinite simple ring.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6a5d41c29069ef9c06f7a9c0a55273961157ea5a

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

Purely infinite simple Leavitt path algebras (2005, background): “[3], [4], [5]) as well as an analytic (see e.”
Socle theory for Leavitt path algebras of arbitrary graphs (2008, background): “The algebraic counterpart of th ese finite CuntzKrieger algebras was considered in [7].”
Graded Rings and Graded Grothendieck Groups (2014, methodology): “The following theorem (first established in [8]) will be used to realise Leavitt path algebras (§1.”
The Leavitt path algebra of a graph (2005, background): “Keywords: path algebra; Leavitt algebra; Cuntz Krieger C*-algebra”
Irreducible representations of Leavitt path algebras (2011, background): “Q/ of Q with coefficients in k is relatively new, which is introduced in [1, 7, 8].”
CATEGORY EQUIVALENCES INVOLVING GRADED MODULES OVER PATH ALGEBRAS OF QUIVERS (2011, background): “It makes use of a construction in [2] that we now recall.”
Leavitt path algebras: the first decade (2014, methodology): “) As mentioned in the Historical Plot Line #1, in work that predates the introduction of the general definition of Leavitt path algebras the four authors of [29] investigated the notion of a fractional skew monoid ring, which in particular situations is denoted A[t+, t−, α].”