Abstract | Let Γ be a cancelation monoid with the neutral element e . Consider a Γ -graded ring A = ⊕ γ ∈ Γ A γ , 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 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 A e hold true for A (at least) in the Γ -graded context. |