Let $A=\bigoplus_{n \ge 0}A_n$ be a positively graded commutative ring. Let $J_n \le A_n$ be subgroups for $j=0,\ldots,N$, satisfying
- $\,\,\,A_n \cdot J_m \subseteq J_{n+m}$ for $n+m \le N$
- $\,\,\,x \cdot y \in J \Rightarrow x \in J$ or $y \in J$ for all homogeneous $x,y \in A$ with $\deg(xy)\le N$
where $J = \bigoplus_{n=0}^NJ_n$.
These two conditions just say that $J$ behaves like a prime ideal in degrees where the products are defined.
Question: Can $J$ be extended to a homogeneous prime ideal, i.e. is there a homogeneous prime ideal $P \subseteq A$ such that $P_n = J_n$ for $n=0,\ldots,N$ ?
Motivation: The question came up to me than I was thinking about this question: Characterization of a finitely graded (almost) domain: If the question above has an affirmative answer and if I'm not mistaken, then the ideals in the linked question are exactly of the form $I= P + R_{> N}$ for a prime ideal $P$. But I think, the question above is interesting in its own right to ask it as a separate question.
