#### Vol. 9, No. 1, 2016

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Completions of reduced local rings with prescribed minimal prime ideals

### Susan Loepp and Byron Perpetua

Vol. 9 (2016), No. 1, 101–118
##### Abstract

Let $T$ be a complete local ring of Krull dimension at least one, and let ${\mathsc{C}}_{1},{\mathsc{C}}_{2},\dots ,{\mathsc{C}}_{m}$ each be countable sets of prime ideals of $T$. We find necessary and sufficient conditions for $T$ to be the completion of a reduced local ring $A$ such that $A$ has exactly $m$ minimal prime ideals ${Q}_{1},{Q}_{2},\dots ,{Q}_{m}$, and such that, for every $i=1,2,\dots ,m$, the set of maximal elements of $\left\{P\in Spec\left(T\right)\mid P\cap A={Q}_{i}\right\}$ is the set ${\mathsc{C}}_{i}$.

##### Keywords
completions of local rings, minimal prime ideals
##### Mathematical Subject Classification 2010
Primary: 13B35, 13F25, 13J05, 13J10