 
Summary: FRACTIONAL SKEW MONOID RINGS
P. ARA, M.A. GONZ ’
ALEZBARROSO, K.R. GOODEARL, AND E. PARDO
Dedicated to the memory of Dmitry Tyukavkin
Abstract. Given an action # 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
## A ## 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 Ggraded
ring S op
## A ## S with the property that, for each s # S, the scomponent 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 # of A. If # is an isomorphism onto a proper corner pAp, we obtain an
analogue of the usual skew Laurent polynomial ring, denoted by A[t + , t  ; #]. 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  ; #]. Finally, mild
and reasonably natural conditions are obtained under which S op
## A## S is a purely infinite
simple ring.
