Let $I$ be a homogeneous ideal in a graded local commutative ring $R$, $S$ be its minimal homogeneous system of generators. So, we know that the cardinality of $S$ is unique as the dimension of the vector space $I/\mathfrak{m}I$, where $\mathfrak{m}$ is the maximal graded ideal of $R$.
My question is the following: Is the degree of each element of $S$ uniquely determined by $I$?
Thank for reading my question.