Overrings of right chain domains BY

For numerous interesting classes C of commutative rings assertions of the following type hold: Let R be a domain in C, T an overring with T Q (R), then T is a quotient ring of R (QR-property; Gilmer [6]) resp. an intersection of quotient rings of R (QQR-property; Gilmer [6]) and T E C. Generalizations to classes of non-commutative domains are also known, e.g. for leftlright principal ideal domains (Brungs [4]) resp. for Bezout domains (Beauregard [3]). However these rings are leftlright symmetric. In general, it can not be expected that similar results hold for rings with only a one-sided structure as the counter example of a right chain domain in Brungs [4] demonstrates. The aim of this paper is the analysis of overrings in exactly this dass of rings with a one-sided structure. In Section 1 we give examples to show that in contrast to the situation for chain domains overrrings T of a right chain domain need not be right chain domains. Further-more these rings need not lie in a chain. Even if the ring R is of rank 1, overrings which are assumed to be right chain domains may have arbitrarily large rank (Section 2). In Section 3 we analyze overrings which contain "bad " elements (Theorem 7, Theorem 8)