Graded Nakayama's lemma

This article is about the statement of a simple but indispensable lemma in commutative algebra
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Statement

Suppose $A$ is a graded ring. Let $A^+$ denote the ideal of all positively graded elements. Then, if $M$ is an $A$-graded module, $A^+M = M \implies M = 0$.

Related results

• Nakayama's lemma