Some remarks on linear structures in right distributive domains

We examine properties of the right ideal structure of right distributive do mains. Right distributive domains R are exactly those rings whose localizat ions at maximal right ideals M are right chain domains R-M. On the one hand , the paper focuses on the question in which way properties of R are carrie d over to R-M and vice versa. We examine the problem under which conditions two-sided ideals of R are again two-sided in the extension R-M (Lemma 2.2) . Further, we observe the relationship between completely prime resp. semip rime ideals of R and the extended ideals in R-M. On the other hand, we prov e in particular that for any maximal right ideal M = R\S-M the right-S-M-sa turation I-[M] of a completely semiprime ideal I subset of or equal to M of R is completely prime (Theorem 2.9). A central role is played by waists of right distributive rings which are right ideals comparable to each other i deal, in particular there exists a largest waist W which is completely prim e. We present a representation theorem in terms of ideals in R,. We apply t hese results to the Jacobson radical J(R) of a right distributive domain R. Illustrative examples are given. 1991 Mathematics Subject Classification: 16L30; 16P50; 16N60.