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 Ri’s from some given star operations ?i on Ri’s. We provide the ultra b-operation and the ultra v-operation. Given an arbitrary star operation ? on the ultraproduct of some integral domains, we pose the problem of whether the restriction of ? to the set of induced ideals is necessarily an ultra star operation. We show that the ultraproduct of integral domains Ri’s is a ?-Pr¨ufer domain if and only if Ri is a ?i-Pr¨ufer domain for U-many i.