ULTRA STAR OPERATIONS ON ULTRA PRODUCT OF INTEGRAL DOMAINS

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 $i$.