Uppers to zero and semistar operations in polynomial rings

摘要

Given a stable semistar operation of finite type star on an integral domain D, we show that it is possible to define in a canonical way a stable semistar operation of finite type [star] on the polynomial ring D[X], such that D is a star-quasi-Prufer domain if and only if each upper to zero in D[X] is a quasi-[star] -maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott [E. Houston, S. Malik, J. Mott, Characterizations of star-multiplication domains, Canad. Math. Bull. 27 (1984) 48-52, Section 2. [17]] in the star operation setting. Moreover, we show that D is a Prufer star-multiplication (respectively, a star-Noetherian; a star-Dedekind) domain if and only if D[X] is a Prufer [star]-multiplication (respectively, a [star]-Noetherian; a [star]-Dedekind) domain. As an application of the techniques introduced here, we obtain a new interpretation of the Gabriel-Popescu localizing systems of finite type on an integral domain D (Problem 45 of [S.T. Chapman, S. Glaz, One hundred problems in commutative ring theory, in: S.T. Chapman, S. Glaz (Eds.), Non-Noetherian Commutative Ring Theory, Kluwer Academic Publishers, 2000, pp. 459-476. [4]]), in terms of multiplicatively closed sets of the polynomial ring D[X]. (c) 2007 Elsevier Inc. All rights reserved.