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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 $S$ ?

2 added 12 characters in body

Let $I$ be a homogeneous ideal in a graded commutative ring $R$, $S$ be its 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 $S$ ?

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# Homogeneous ideal and its system of generators

Let $I$ be a ideal in a graded commutative ring $R$, $S$ be its 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 $S$ ?