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arXiv · 1108.0258

On Monoid Graded Local Rings

Huishi Li
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Abstract

Let $Γ$ be a cancelation monoid with the neutral element $e$. Consider a $Γ$-graded ring $A=\oplus_{γ\inΓ}A_γ$, 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 $\mathfrak{M}$ 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.

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Huishi Li. 2011-08-18. On Monoid Graded Local Rings. https://arxiv.org/abs/1108.0258

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