  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
$S^{\rm op} * A * T$ 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 $G$-graded ring $S^{\rm op} * A * S$ with
the property that, for each $s\in S$, the $s$-component contains a left
invertible element and the $s^{-1}$-component contains a right invertible
element. In the most basic case, where $G$ is the additive group of integers
and $S=T$ is the submonoid of nonnegative integers, 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 $S^{\rm op} * A * S$ is a purely infinite simple ring.
