Power series over Noetherian domains, Nagata rings, and Kronecker function rings

JOURNAL OF ALGEBRA

卷:468

Article

Chang, Gyu Whan¹ 

[1] Incheon Natl Univ, Dept Math Educ, Inchon 22012, South Korea

关键词: Power series ring over a Noetherian domain;    Star operation;    Nagata ring;    Kronecker function ring;   

DOI  :  10.1016/j.jalgebra.2016.07.040

来源: Elsevier

【 摘 要 】

Let D be a Noetherian domain, * be a star operation on D, X be an indeterminate over D, D[X] be the power series ring over D, c(f) be the ideal of D generated by the coefficients of f is an element of D[X], and N-* = {f is an element of D[X] vertical bar c(f)* = D}. Moreover, if * is e.a.b., then we let Kr((D,*)) = {f/g vertical bar f, g is an element of D[X], 0 not equal g, and c(f) subset of c(g)*}. In this paper, we show that N-* is a saturated multiplicative set and Kr((D, *)) is a Bezout domain. We then study some ring-theoretic properties of D[X](N). and Kr((D, *)). For example, we prove that every invertible ideal of D[X](N*), is principal; dim (Kr((D, b))) = dim(v) (D); and if V is a valuation overring of D, then (V) over cap = {f/g vertical bar f, g is an element of D[X], g not equal 0, and c(f)V subset of c(g)V} is a valuation overring of D[X]; and Kr((D, *)) = boolean AND{(V) over cap vertical bar V is a *-valuation overring of D}. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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