Nakayama lemma

For a compact continuous module over the Iwasawa algebra of a Zp-extension, lifts of a basis of $M/(p,T)M$ generate $M$. Their generated image is compact. The quotient $Q$ satisfies $Q=(p,T)Q$; in every finite continuous quotient, $(p,T)$ acts nilpotently, since the group action is pro-$p$. Thus all finite quotients of $Q$ vanish, and compactness gives $Q=0$.

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 $M$.