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Let $G$ be an abelian group and $R$ be a $G$-graded ring.

Question $1$:

Is there a map $\phi:\mathbb{N}\rightarrow\mathbb{N}$ such that for every $n\in \mathbb{N}$ and any homogeneous ideal $I$ of $R$ generated by $n$ elements, $I$ can be generated by $\phi(n)$ homogeneous elements ?

If we denote by $\mu_R(I)$ the minimal number of homogeneous generators of $I$, this is equivalent to the following:

Is for every $n\in \mathbb{N}$, $Sup_I \ \ \mu_R(I) < \infty$, where $I$ runs over the set of homogeneous ideal of $R$ generated by $n$ elements ?

Question $2$:

We consider the following assertions.

$(1)$ There is a map $\psi:\mathbb{N}\rightarrow\mathbb{N}$ such that for every $n\in\mathbb{N}$, and any homogeneous ideal $I$ of $R$ generated by $n$ homogeneous elements and any homogeneous element $a\in R$, $(I:a)$ can be generated by $\psi(n)$ homogeneous elements.

$(2)$ There is a map $\phi:\mathbb{N}\rightarrow\mathbb{N}$ such that for every $n\in\mathbb{N}$, and any homogeneous ideal $I$ of $R$ generated by $n$ homogeneous elements and any homogeneous element $a\in R$, $(I:a)$ can be generated by $\phi(n)$ elements.

Obviously, $(1)\Rightarrow(2)$. But I have been unable to determine if the converse is true in general.

Thank you very much.

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No, not in general. For instance, consider the $\mathbb{Z}$-graded ring $R$ with $R_n=\mathbb{Z}/(n)$ for each $n$ where all products of homogenenous elements of nonzero degree are $0$. Given any pairwise coprime integers $n_1,\dots,n_k$, consider the element $x$ which is $1$ in degrees $n_1,\dots,n_k$ and $0$ in all other degrees. Then $x$ generates a homogeneous ideal, since each homogeneous part of $x$ can be written as $mx$ for some $m\in\mathbb{Z}$ (choose $m$ which is $1$ mod $n_i$ and $0$ mod $n_j$ for all $j\neq i$). But clearly $(x)$ cannot be generated by fewer than $k$ homogeneous elements. So, there are principal homogeneous ideals in $R$ which require arbitrarily large numbers of homogeneous generators.

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  • $\begingroup$ Thank you very much Mr. Eric for your time. This is an interesting example. $\endgroup$ – A. R. Mar 13 at 0:41

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