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Contents of polynomials and invertibility. (English) Zbl 0705.13005

Let \(D\) be a commutative integral domain with identity, \(K\) its quotient field and \(D[x]\) the ring of polynomials in one variable. The \(D\)-content \(A_f\) of \(f\in D[x]\) is defined to be the fractional ideal of \(D\) generated by the coefficients of \(f\). As is well-known \(A_{fg}\subseteq A_f A_g\) always holds.
In this paper some results are derived which connect that notion with properties of \(D\). For example, Theorem 1.5 says (among others), that the following statements are equivalent:
(1) \(D\) is integrally closed in \(K\);
(2) \(S=\{f\in K[x]\), \(A_ f\) is invertible} is a saturated multiplicative system in \(K[x]\);
(3) If \(f,g\in K[x]\) with \(A_{fg}\subseteq D\) then \(A_f A_g\subseteq D\).
Theorem 1.7 states that the integral closure \(\bar D\) of \(D\) in \(K\) is a Prüfer domain iff the saturation of \(S\cap D[x]\) in \(\bar D[x]\) equals \(D[x]\setminus \{0\}\) iff each \(\alpha\in K\), \(\alpha\neq 0\), satisfies an \(f\in D[x]\) such that \(A_fD\) is invertible. Also a simple condition in terms of \(A_f\) is given which forces a Krull domain to be Dedekind. Parallel results are obtained for polynomials \(f\) with \(v\)-invertible \(A_f\), which means \(((A_fA_f^{-1})^{-1})^{-1}=D\).

MSC:

13B25 Polynomials over commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13B22 Integral closure of commutative rings and ideals
13G05 Integral domains
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References:

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