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Abstract

Let $R$ be a commutative ring. If $A\subseteq R$ is an ideal and $\mathcal F$ is a monoidal semifilter of ideals in $R$, we say that a prime ideal $P$ is a realization of $(A,\mathcal F)$ if $P\supseteq A$ and $P\notin \mathcal F$. We give “if and only if” conditions for the existence of a realization of a family $\{(A_t,\mathcal F_t)\}_{t\in T}$ of such pairs indexed by a finite rooted tree $T$. We also apply our results to trees of prime ideals outside a given monoidal semifilter in a tensor product of algebras.