Terms Of Ideals Research Articles

Some remarks on linear structures in right distributive domains

We examine properties of the right ideal structure of right distributive domains. Right distributive domains R are exactly those rings whose localizations 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 carried 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. semiprime ideals of R and the extended ideals in R M. On the other hand, we prove in particular that for any maximal right ideal M^ R)S M the right-S M -saturation I %M& of a completely semiprime ideal I 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 ideal, in particular there exists a largest waist W which is completely prime. We present a representation theorem in terms ofideals in R W. We applythese results to the Jacobsonradical J(R) ofa rightdistributive domain R. Illustrative examples are given.