Let be an integral domain with quotient field , be an indeterminate over , be the power series ring over , and be the ideal of generated by the coefficients of . We will say that a star operation on is a c-star operation if (i) for all and (ii) implies for all nonzero fractional ideals of . Assume that admits a c-star operation , and let . Among other things, we show that is a Bézout domain, is completely integrally closed, the -operation on is a c-star operation, and is a completely integrally closed Bézout domain. We also show that if is a rank-one valuation domain, then the -operation on is a c-star operation, is a rank-one valuation domain, and is a DVR if and only if is a DVR. Using this result, we show that if is a generalized Krull domain, then is a one-dimensional generalized Krull domain.
"Kronecker function rings and power series rings." J. Commut. Algebra 12 (1) 27 - 51, Spring 2020. https://doi.org/10.1216/jca.2020.12.27