This topic suggested me the following question:
If $R$ is a commutative graded ring and $F$ a graded $R$-module which is free, then can we conclude that $F$ has a homogeneous basis (that is, a basis consisting of homogeneous elements)?
In general the answer is negative, and such an example can be found in Nastasescu, Van Oystaeyen, Methods of Graded Rings, page 21. But their example is not quite usual, and
I'd like to know if however the property holds for positively graded $K$-algebras, for example.
If not, then maybe someone can provide a generic example when the property holds.