Gyu Whan Chang. Source: Journal of Commutative Algebra, Volume 12, Number 1, 27--51.Abstract:
Let [math] be an integral domain with quotient field [math] , [math] be an indeterminate over [math] , [math] be the power series ring over [math] , and [math] be the ideal of [math] generated by the coefficients of [math] . We will say that a star operation [math] on [math] is a c-star operation if (i) [math] for all [math] and (ii) [math] implies [math] for all nonzero fractional ideals [math] of [math] . Assume that [math] admits a c-star operation [math] , and let [math] . Among other things, we show that [math] is a Bézout domain, [math] is completely integrally closed, the [math] -operation on [math] is a c-star operation, and [math] is a completely integrally closed Bézout domain. We also show that if [math] is a rank-one valuation domain, then the [math] -operation on [math] is a c-star operation, [math] is a rank-one valuation domain, and [math] is a DVR if and only if [math] is a DVR. Using this result, we show that if [math] is a generalized Krull domain, then [math] is a one-dimensional generalized Krull domain.