 Graded Nakayama lemma

For a positively graded ring with degree-zero part a field, put $R_{+}={⨁}_{d>0}{R}_d$. 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 $M$.