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Can a ideal-like subset that is prime in low degrees be extended to a prime ideal ?

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 $n+m \le N$ $\,\,\ …