On Monoid Graded Local Rings

Abstract

Let $\Gamma$ be a cancelation monoid with the neutral element $e$. Consider a $\Gamma$-graded ring $A=\oplus_{\gamma\in\Gamma}A_{\gamma}$, which is not necessarily commutative. It is proved that $A_e$, the degree-$e$ part of $A$, is a local ring in the classical sense if and only if the graded two-sided ideal $\mathfrak{M}$ of $A$ generated by all non-invertible homogeneous elements is a proper ideal. Defining a $\Gamma$-graded local ring $A$ in terms of this equivalence, it is proved that any two minimal homogeneous generating sets of a finitely generated $\Gamma$-graded $A$-module have the same number of generators, and furthermore, that most of the basic homological properties of the local ring $A_e$ hold true for $A$ (at least) in the $\Gamma$-graded context.