Rings, Modules, and Closure Operations by Jesse Elliott

Author:Jesse Elliott
Language: eng
Format: epub, pdf
ISBN: 9783030244019
Publisher: Springer International Publishing

Next, using similar methods, we characterize the -Prüfer rings for any semistar operation .

Definition 3.8.2.

Let be a semistar operation on a ring R. The ring R has enough regular -prime ideals if every proper regular -closed ideal of R is contained in a -prime ideal of R (or equivalently, if every regular ideal I of R such that is contained in some -prime ideal of R).

By Lemma 3.2.8, if is of r-finite type, then R has enough regular -prime ideals.

Proposition 3.8.3.

Let R be a ring with total quotient ring K, and let be a semistar operation on R. Each of the following conditions implies the next.

(1)R is -Prüfer.

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