The global dimension of a trace monoid ring

The article is devoted to the global dimension of a R-monoid ring RM(E,I) of a free partially commutative monoid. We prove that if the supremum of numbers of distinct pairwise commuting elements in E is equal to n, then gl dim RM(E,I)=n+gl dim R. Moreover, we get a generalized formula for the global dimensions of the categories of M(E,I)-objects in Abelian categories.