ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS

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International Electronic Journal of Algebra
Volume:27 Issue:27
ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS

ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS

Authors : Olivier A HEUBOKWEGNA

Pages : 206-217

Doi:10.24330/ieja.663071

View : 31 | Download : 12

Publication Date : 2020-01-07

Article Type : Research Paper

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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