It's known that polynomial rings $R[x_1,...,x_n]$ inherit some properties of a base ring $R$. For example, (due to the wikipedia article on polynomial rings) if $R$ is an integral domain, then so is $R[x_1,...,x_n]$ and if $R$ is a UFD, then so is $R[x_1,...,x_n]$.
General monoid rings are generalizations of the notion of a polynomial rings in finitely many variables. For example, if $X$ is an infinite set, then we can form a "polynomial ring" in $|X|$ many variables as $R[M]$ where $M$ is the free commutative monoid on $X$. Of course, if $X = \{x_1,...,x_n\}$, then this construction yields a usual polynomial ring $R[x_1,...,x_n]$. On the other hand, $R[M]$ is a free algebra on $X$ for the free (non-commutative) monoid $M$ on $X$.
My question is: what properties of $R$ are preserved by $R[M]$ for a monoid $M$? For instance, is being an integral domain is preserved? I would also like some references on the matter.
No, being an integral domain is not preserved; for example, if $M$ is the free idempotent (the free monoid on an element $m$ satisfying $m^2 = m$) then $R[M] \cong R[m]/(m^2 - m) \cong R \times R$. I don't know anything useful to say about what is preserved. The monoid ring construction is very general.