Graded Nakayama lemma
= Graded Nakayama lemma
For a positively <graded ring> with degree-zero part a <field>, put $R_+=\bigoplus_{d>0}R_d$. A bounded-below <graded module> $M$ with $M=R_+M$ is zero: a nonzero homogeneous element of least degree would have to be a sum of positive-degree coefficients times elements of lower degree. Hence homogeneous lifts of a basis of $M/R_+M$ generate $M$.