Fractional skew monoid rings

Given an action alpha 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 (*alpha) A (*alpha) 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-1 S, we obtain a G-graded ring S-op (*alpha) A (*alpha) S with the property that, for each s epsilon 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 alpha of A. If alpha is an isomorphism onto a proper corner pAp, we obtain an analogue of the usual skew Laurent polynomial ring, denoted by A [t(+), t_; alpha]. 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_; alpha]. Finally, mild and reasonably natural conditions are obtained under which S-op (*alpha) A (*alpha) S is a purely infinite simple ring. (C) 2004 Elsevier Inc. All rights reserved.