On Monoid Graded Local Rings

Let $\mathrm{\Gamma }$ be a cancelation monoid with the neutral element $e$. Consider a $\mathrm{\Gamma }$-graded ring $A={\oplus }_{\gamma \in \mathrm{\Gamma }}{A}_{\gamma }$, 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 $\mathrm{\Gamma }$-graded local ring $A$ in terms of this equivalence, it is proved that any two minimal homogeneous generating sets of a finitely generated $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }$-graded context.