I don't have my hands on the book of Nastasescu and Van Oystaeyen on graded rings, which I believe contains this result, but here is a reference to a paper by H. Huishi Li, "On Monoid Graded Local Rings", where the author proves, in section 2, that in a $\Gamma$-graded ring, $A$, where $A=\bigoplus_{\gamma\in\Gamma}A_{\gamma},$$where $\Gamma$ is a cancellation monoid with neutral element $e$, one has that $A_e$ is a local ring if and only if the graded two-sided ideal generated by homogeneous non-units is proper.
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I don't have my hands on the book of Nastasescu and Van Oystaeyen on graded rings, which I believe contains this result, but here is a reference to a paper by H. Li, "On Monoid Graded Local Rings", where the author proves, in section 2, that in a $\Gamma$-graded ring, $A$, where $\Gamma$ is a cancellation monoid with neutral element $e$, one has that $A_e$ is a local ring if and only if the graded two-sided ideal generated by homogeneous non-units is proper. |
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