FORMALLY INTEGRALLY CLOSED DOMAINS AND THE RINGS R((X)) AND R{{X}}

Abstract

Let R be an integral domain. For f is an element of R[X] let A(f) be the ideal of R generated by the coefficients of f. We define R to be formally integrally closed double left right arrow (A(fg))(t) = (A(f)A(g)), for all nonzero f, g is an element of R[X]. Examples of formally integrally closed domains include locally finite intersections of one-dimensional Prufer domains (e.g., Krull domains and one-dimensional Prufer domains). We study the rings R((X)) = R[X](N) and R{{X}} = R[X](Nt) where N = (f is an element of R[X]A(f) = R) and N-t = (f is an element of R[X](A(f)) = R). We show thar R is a Krull domain (resp., Dedekind domain) double left right arrow R{{X}}(resp., R((X))) is a Krull domain (resp., Dedekind domain) double left right arrow R{{X}}(resp., R((X))) is a Euclidean domain double left right arrow every(principal) ideal of R{{X}} (resp., R((X))) is extended from R double left right arrow R is formally integrally closed and every prime ideal of R{{X}} (resp., R((X))) is extended from R. (C) 1998 Academic Press.