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The construction D + XDS[X]

1978; Elsevier BV; Volume: 53; Issue: 2 Linguagem: Inglês

10.1016/0021-8693(78)90289-2

1090-266X

Douglas Costa, Joe L. Mott, Muhammad Zafrullah,

Algebraic Geometry and Number Theory

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 cases we 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