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., .
Read more about this topic: Nakayama Lemma
Famous quotes containing the words graded and/or version:
“I don’t want to be graded on a curve.”
—Mary Carillo (b. 1957)
“Truth cannot be defined or tested by agreement with ‘the world’; for not only do truths differ for different worlds but the nature of agreement between a world apart from it is notoriously nebulous. Rather—speaking loosely and without trying to answer either Pilate’s question or Tarski’s—a version is to be taken to be true when it offends no unyielding beliefs and none of its own precepts.”
—Nelson Goodman (b. 1906)