Halter-Koch, Franz Kronecker function rings and generalized integral closures. (English) Zbl 1073.13507 Commun. Algebra 31, No. 1, 45-59 (2003). 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. Cited in 6 ReviewsCited in 21 Documents 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 Keywords:Kronecker function rings; ideal system; star operation; algebraic field extensions × Cite Format Result Cite Review PDF 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.