Graded Version
There is also a graded version of Nakayama's lemma. Let R be a graded ring (over the integers), and let denote the ideal generated by positively graded elements. Then if M is a graded module over R for which for i sufficiently negative (in particular, if M is finitely generated and R does not contain elements of negative degree) such that, then . Of particular importance is the case that R is a polynomial ring with the standard grading, and M is a finitely generated module.
The proof is much easier than in the ungraded case: taking i to be the least integer such that, we see that does not appear in, so either, or such an i does not exist, i.e., .
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