ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS

Year 2020, Volume: 27 Issue: 27, 206 - 217, 07.01.2020

Olivier A. Heubo-kwegna

https://doi.org/10.24330/ieja.663071

Abstract

We introduce the notion of ultra star operation on ultraproduct of integral domains as a map from the set of induced ideals into the set of induced ideals satisfying the traditional properties of star operations. A case of special interest is the construction of an ultra star operation on the ultraproduct of integral domains $R_i$'s from some given star operations $\star_i$ on $R_i$'s. We provide the ultra $b$-operation and the ultra $v$-operation. Given an arbitrary star operation $\star$ on the ultraproduct of some integral domains, we pose the problem of whether the restriction of $\star$ to the set of induced ideals is necessarily an ultra star operation. We show that the ultraproduct of integral domains $R_i$'s is a $\star$-Pr\"{u}fer domain if and only if $R_i$ is a $\star_i$-Pr\"{u}fer domain for $\mathcal{U}$-many $i$.

Keywords

Star operation, ultraproduct of domains, Prüfer domain

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