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**\(P\)-orderings and polynomial functions on arbitrary subsets of Dedekind rings.**
*(English)*
Zbl 0899.13022

This paper investigates polynomial mappings on subsets of finite principal ideal rings and the ring \(\operatorname{Int}(X,D)\) of polynomials which map a given subset \(X\) of a Dedekind domain \(D\) to itself. For this the concept of \(P\)-ordering is introduced in the class of Dedekind rings which in this paper means a noetherian, locally principal ring wherein all non-zero prime ideals are maximal. Here are samples of some interesting results:

Let \(R\) be a Dedekind ring, \(P\) a prime ideal and \(X\) an arbitrary subset of \(R\). Then any two \(P\)-orderings of \(X\) result in the same associated \(P\)-sequence. A complete elementwise description of the ideal of polynomials in \(R[X]\) where \(R\) is any non-trivial quotient of a Dedekind domain vanishing on a given subset \(X\) of \(R\). Theorem 5 gives the number of polynomial functions from \(X\) to \(R\) and theorem 6 gives a congruence criterion for polynomial representability of a function from \(X\) to \(R\). In case \(R\) is a finite principal ideal ring and \(X\) is a subset of it, then all functions from \(X\) to \(R\) are polynomial functions if and only if for each prime ideal \(P\) of \(R\), no two elements of \(X\) are congruent modulo \(P\). Existence of a regular basis for the pair \((X,D)\) (if the ring \(\operatorname{Int}(X,D)\) is a free \(D\)-module with \(D\)-basis consisting of exactly one polynomial of each degree) and an explicit construction of a regular basis whenever it exists are given in theorem 14. In section 6, many of the classical results on polynomial functions are derived as consequences of the theorems established here. In the concluding section some open questions are mentioned.

The paper is well written and has several interesting examples.

Let \(R\) be a Dedekind ring, \(P\) a prime ideal and \(X\) an arbitrary subset of \(R\). Then any two \(P\)-orderings of \(X\) result in the same associated \(P\)-sequence. A complete elementwise description of the ideal of polynomials in \(R[X]\) where \(R\) is any non-trivial quotient of a Dedekind domain vanishing on a given subset \(X\) of \(R\). Theorem 5 gives the number of polynomial functions from \(X\) to \(R\) and theorem 6 gives a congruence criterion for polynomial representability of a function from \(X\) to \(R\). In case \(R\) is a finite principal ideal ring and \(X\) is a subset of it, then all functions from \(X\) to \(R\) are polynomial functions if and only if for each prime ideal \(P\) of \(R\), no two elements of \(X\) are congruent modulo \(P\). Existence of a regular basis for the pair \((X,D)\) (if the ring \(\operatorname{Int}(X,D)\) is a free \(D\)-module with \(D\)-basis consisting of exactly one polynomial of each degree) and an explicit construction of a regular basis whenever it exists are given in theorem 14. In section 6, many of the classical results on polynomial functions are derived as consequences of the theorems established here. In the concluding section some open questions are mentioned.

The paper is well written and has several interesting examples.

Reviewer: N. Sankaran (Bangalore)

### MSC:

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |

13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |

13F30 | Valuation rings |

13M10 | Polynomials and finite commutative rings |