Abstract

Introduction. All in this paper are commutative with identity element, and all subrings possess the identity element of the containing ring. An overring of an integral domain denotes here a subring of the quotient field containing the given ring. We assume the reader conversant with the elements of commutative (Noetherian) ring theory presented in the textbook of Zariski and Samuel [8], especially in those passages dealing with quotient rings, integral extensions, and the elementary theorems on valuation rings. Let that book serve as a general reference for the well-known definitions and results we introduce here with little or no comment. It is a fact that every overring of a Priifer ring (in the Noetherian case, ring) is also a Priufer ring, and, a fortiori, integrally closed. Further, for some Priifer (e.g., the ring of integers) all overrings are of quotients. Our purpose here is the study of with such properties, namely: P-domains, integral domains for which all overrings are integrally closed, and Q-domains, integral domains for which all overrings are of quotients. In ?1 we see that P-domains are indeed Priifer and that Q-domains are a special class of Pruifer (in the Noetherian case, exactly those with torsion class group). In ?2 and ?3 we consider P-rings and Q-rings in the context of with divisors of zero. ?3, which deals exclusively with Noetherian rings, is essentially a consideration of the notion Dedekind rings in the presence of zero-divisors. ?2 treats a special case of (sometimes) non-Noetherian rings. Since the results of ?2 and ?3 include as special cases many of the results of ?1, the reader is due an explanation of this uneconomical style of exposition. The excuse is twofold: (1) The consideration of integral domains contains the basic ideas of this study and serves to motivate the more general considerations. (2) The essentially simple ideas involved here tend to become obscured by the technicalities required in the presence of zero-divisors. We hope that this choice of exposition will produce a gain in clarity sufficient to compensate for the increased length. We shall attempt to avoid duplication of arguments by only

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