On monoid graded local rings

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