Monoid ring

Abstract: For a monoid M, we introduce M-Armendariz rings, which are generalizations of Armendariz rings; and we investigate their properties. Every reduced ring is M-Armendariz for any unique product monoid M. We show that if R is a reduced and M-Armendariz ring, then R is M × N-Armendariz, where N is a unique product monoid. It is also shown that a finitely generated Abelian group G is torsion free if and only if there exists a ring R such that R is G-Armendariz. Moreover, we study the relationship between the Baerness and the PP-property of a ring R and those of the monoid ring R[M] in case R is M-Armendariz.

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 $S^{\rm op} * A * T$ 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.