Let $I$ be a homogeneous ideal in a graded commutative ring $R$, $S$ be its minimal system of generators.

What is the conclusion that we can say about the element in $S$ ? Is the cardinality of $S$ uniquely determined by $I$ ?

In the book Commutative ring theory of Matsumura, theorem 2.3, page 8 there is a theorem for local ring which we can deduce that the number of generator is unique. So, is there a version of that theorem for the graded ring ?

What about the degree of generator in $S$ ?

Graded Nakayama's Lemma. Everything you want is true if you assume that $R$ is $\mathbb{N}$-graded with $R_0$ a field. In particular the same argument goes through from the local case (page 9 of Matsumura). $\endgroup$