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Kronecker function rings and generalized integral closures. (English) Zbl 1073.13507

From the paper: Let \(D\) be an integral domain, \(K\) a field containing \(D\) and \(K(X)\) a rational function field over \(K\).
Definition. By a \(K\)-function ring we mean a subring \(R\subset K(X)\) with the following properties:
\(X\in R^\times\).
For all \(f\in K[X]\), we have \(f(0)\in fR\).
If \(D= R\cap K\), then \(R\) is called a function ring of \(D\).
In the paper under review, the author establishes the connection between \(K\)-function rings and the classical Kronecker function rings, provides an axiomatic concept of Kronecker function rings and applies it to associate a Kronecker function ring to any integral domain \(D\) and any ideal system (star operation) on \(D\). He investigates its behavior in algebraic field extensions and its connection with the defining valuation domains.

MSC:

13G05 Integral domains
13B22 Integral closure of commutative rings and ideals
13A15 Ideals and multiplicative ideal theory in commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
11R58 Arithmetic theory of algebraic function fields
Full Text: DOI

References:

[1] Gilmer R., Multiplicative Ideal Theory (1972) · Zbl 0248.13001
[2] Halter-Koch F., Factorization in Integral Domains (1997) · Zbl 0882.13027
[3] Halter-Koch F., Ideal Systems (1998) · Zbl 0953.13001
[4] DOI: 10.1007/BF01180441 · Zbl 0015.00203 · doi:10.1007/BF01180441
[5] DOI: 10.1007/BF01180447 · Zbl 0015.24501 · doi:10.1007/BF01180447
[6] DOI: 10.1007/BF02678019 · Zbl 0901.13001 · doi:10.1007/BF02678019
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