Fractional skew monoid rings

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 ∗ αA∗ αT 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 −1 S, we obtain a G-graded ring S op ∗ αA∗ αS 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 ∗ αA∗ αS is a purely infinite simple ring.