Coprime packedness and set theoretic complete intersections of ideals in polynomial rings

Erdoǧdu, V.

doi:10.1090/S0002-9939-04-07438-6

Coprime packedness and set theoretic complete intersections of ideals in polynomial rings
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by V. Erdoǧdu PDF

Proc. Amer. Math. Soc. 132 (2004), 3467-3471 Request permission

Abstract:

A ring $R$ is said to be coprimely packed if whenever $I$ is an ideal of $R$ and $S$ is a set of maximal ideals of $R$ with $I\subseteq \bigcup \{M\in S\}$, then $I\subseteq M$ for some $M\in S$. Let $R$ be a ring and $R\langle X\rangle$ be the localization of $R[X]$ at its set of monic polynomials. We prove that if $R$ is a Noetherian normal domain, then the ring $R\langle X\rangle$ is coprimely packed if and only if $R$ is a Dedekind domain with torsion ideal class group. Moreover, this is also equivalent to the condition that each proper prime ideal of $R[X]$ is a set theoretic complete intersection. A similar result is also proved when $R$ is either a Noetherian arithmetical ring or a Bézout domain of dimension one.

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