On going down for simple overrings
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- by David E. Dobbs PDF
- Proc. Amer. Math. Soc. 39 (1973), 515-519 Request permission
Abstract:
Let $R$ be an integral domain with quotient field $K$. If $R$ is Noetherian: then the Krull dimension of $R$ is at most $1 \Leftrightarrow$for all overrings $S$ of $R$, $R \subset S$ satisfies going down. $R$ is Dedekind ${\text {(resp}}{\text {., PID)}} \Leftrightarrow R$ is Krull (resp., UFD) and, for all $u \in K,R \subset R[u]$ satisfies going down. $R$ is Prüfer$\Leftrightarrow R$ is integrally closed, every intersection of two principal ideals of $R$ is finitely generated, and $R \subset R[u]$ satisfies going down for all $u \in K$.References
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Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 39 (1973), 515-519
- MSC: Primary 13B20
- DOI: https://doi.org/10.1090/S0002-9939-1973-0417152-3
- MathSciNet review: 0417152