JOURNAL OF ALGEBRA 53, 423-439 (1978) The Construction D + XD,[X] DOUGLAS COSTA University of Virginia, Charlottesville, Virginia 22901 JOE L. MOTT Florida State University, Tallahassee, Florida 32306 AND MUHAMMAD ZAFRULLAH I7 Strathearn Road, London, S. W. 19, England Communicated by P. M. Cohn Received July 14, 1975 If D is a commutative integral domain and S is a multiplicative system in D, then Tfs) = D + XD,[X] is the subring of the polynomial ring D,[X] con- sisting of those polynomials with constant term in D. In the special case where S = D* = D\(O), we omit the superscript and let T denote the ring D + XK[XJ, where K is the quotient field of D. Since Tfs) is the direct limit of the rings D[X/s], where s E S, we can conclude that many properties hold in T@) because these properties are preserved by taking polynomial ring extensions and direct limits. Moreover, the ring Tcs) is the symmetric algebra S,(D,) of D, considered as a D-module. In addition, Ds[Xj is a quotient ring of Tts) with respect to S; in fact, in the terminology of [lo], Tfs) is the composite of D and D,[iYj over the ideal XDJX]. (The most familiar of the composite constructions is the so-called D + M construction [l], where generally M is the maximal ideal of a valuation ring.) The ring T ts), therefore, provides a test case for many questions about direct limits, symmetric algebras, and composites. The state of our knowledge of T is considerably more advanced than that of VJ; generally speaking, we often show that a property holds in T if and only if it holds in D. In other caseswe show that Tcs) does not have a given property if D, # K. For example, if T(S) is a Priifer domain, then D,[xJ is a Prtifer domain and D, is therefore equal to K. We show that T is Priifer (Bezout) if and only if D is Prtifer (Bezout). Yet Tts) is a GCD-domain if D is a GCD- domain and the greatest common divisor of d and X exists in T(S) for each 423 0021-8693/78/0532-0423$02.00/O Copyright 0 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.