Integer-valued polynomials and Prüfer \(v\)-multiplication domains. (English) Zbl 0961.13012
If \(D\) is a domain with quotient field \(K\) then the ring of integer valued polynomials over \(D\) is \(\text{Int}(D)=:\{f \in K[X] \mid f(D) \subseteq D\}\).
This paper is devoted to relating certain properties of \(\text{Int}(D)\) to those of \(D\). A domain, \(D\), is Prüfer if for each prime ideal, \(P\), \(D_P\) is a valuation domain. It is a Prüfer \(v\)-multiplication domain (P\(v\)MD) if \(D_P\) is a valuation domain for each \(t\)-prime ideal, \(P\), i.e., prime ideal which is equal to its \(t\)-closure. K. A. Loper [Proc. Am. Math. Soc. 126, No. 3, 657-660 (1998; Zbl 0887.13010)] characterized those domains for which \(\text{Int}(D)\) is Prüfer.
The current paper does the same for P\(v\)MD’s. The authors partition the collection of prime ideals, \(\text{spec}(D)\), into the set \(\Delta_0\) of int-prime ideals, \(P\), for which \(\text{Int}(D) \not\subseteq D_P[X]\) and \(\Delta_1\), of polynomial prime ideals, \(P\), for which \(\text{Int}(D) \subseteq D_P[X]\). The main result is that \(\text{Int}(D)\) is a P\(v\)MD if and only if \(D\) is a P\(v\)MD, each element of \(\Delta_0\) is a height \(1\) (maximal) ideal and each element of \(\Delta_1\) which is a \(t\)-prime ideal contains a finitely generated ideal not contained in any element of \(\Delta_0\). If we let \(D_0 = \bigcap_{\Delta_0} D_P\) and \(D_1 = \bigcap_{\Delta_1} D_p\) then it is also shown that \(\text{Int}(D) = \text{Int}(D_0) \cap D_1 [X]\) and that if \(\text{Int}(D)\) is a P\(v\)MD, then \(\text{Int}(D_0)\) is Prüfer.
This paper is devoted to relating certain properties of \(\text{Int}(D)\) to those of \(D\). A domain, \(D\), is Prüfer if for each prime ideal, \(P\), \(D_P\) is a valuation domain. It is a Prüfer \(v\)-multiplication domain (P\(v\)MD) if \(D_P\) is a valuation domain for each \(t\)-prime ideal, \(P\), i.e., prime ideal which is equal to its \(t\)-closure. K. A. Loper [Proc. Am. Math. Soc. 126, No. 3, 657-660 (1998; Zbl 0887.13010)] characterized those domains for which \(\text{Int}(D)\) is Prüfer.
The current paper does the same for P\(v\)MD’s. The authors partition the collection of prime ideals, \(\text{spec}(D)\), into the set \(\Delta_0\) of int-prime ideals, \(P\), for which \(\text{Int}(D) \not\subseteq D_P[X]\) and \(\Delta_1\), of polynomial prime ideals, \(P\), for which \(\text{Int}(D) \subseteq D_P[X]\). The main result is that \(\text{Int}(D)\) is a P\(v\)MD if and only if \(D\) is a P\(v\)MD, each element of \(\Delta_0\) is a height \(1\) (maximal) ideal and each element of \(\Delta_1\) which is a \(t\)-prime ideal contains a finitely generated ideal not contained in any element of \(\Delta_0\). If we let \(D_0 = \bigcap_{\Delta_0} D_P\) and \(D_1 = \bigcap_{\Delta_1} D_p\) then it is also shown that \(\text{Int}(D) = \text{Int}(D_0) \cap D_1 [X]\) and that if \(\text{Int}(D)\) is a P\(v\)MD, then \(\text{Int}(D_0)\) is Prüfer.
Reviewer: Keith Johnson (Halifax)
MSC:
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
| 13F05 | Dedekind, Prüfer, Krull and Mori rings and their generalizations |
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
Citations:
Zbl 0887.13010References:
| [1] | Cahen, P.-J.; Chabert, J.-L., Integer-Valued Polynomials. Integer-Valued Polynomials, Amer. Math. Soc. Surveys and Monographs, 58 (1997), American Mathematical Society: American Mathematical Society Providence · Zbl 0884.13010 |
| [3] | Chabert, J.-L., Integer-valued polynomials, Prüfer domains and localization, Proc. Amer. Math. Soc., 118, 1061-1073 (1993) · Zbl 0781.13014 |
| [4] | Gilmer, R., Multiplicative Ideal Theory (1972), Dekker: Dekker New York · Zbl 0248.13001 |
| [5] | Gilmer, R., Prüfer domains and rings of integer-valued polynomials, J. Algebra, 129, 502-517 (1990) · Zbl 0689.13009 |
| [6] | Griffin, M., Some results on \(v\)-multiplication rings, Canad. J. Math., 10, 710-722 (1967) · Zbl 0148.26701 |
| [7] | Hasse, H., Zwei Existenztheoremen über algebraische Zahlkörper, Math. Ann., 95, 229-238 (1925) · JFM 51.0141.04 |
| [8] | Heinzer, W.; Ohm, J., An essential ring which is not a \(v\)-multiplication ring, Canad. J. Math., 25, 856-861 (1973) · Zbl 0258.13001 |
| [9] | Houston, E. G.; Zafrullah, M., Integral domains in which each \(t\)-ideal is divisorial, Michigan Math. J., 5, 291-300 (1988) · Zbl 0675.13001 |
| [10] | Jaffard, P., Les Systèmes d’Ideaux (1960), Dunod: Dunod Paris · Zbl 0101.27502 |
| [11] | Kang, B. G., Prüfer \(v\)-multiplication domains and the ring \(R[X]_{ N_v } \), J. Algebra, 123, 151-170 (1989) · Zbl 0668.13002 |
| [12] | Kang, B. G., Some question about Prüfer \(v\)-multiplication domains, Comm. Algebra, 17, 553-564 (1989) · Zbl 0674.13001 |
| [13] | Loper, A., Sequence domains and integer-valued polynomials, J. Pure Appl. Algebra, 119, 185-210 (1997) · Zbl 0960.13005 |
| [15] | Rush, D. E., The condition Int \((R)\)⊆\(R_S}[X]\) and Int(\(R_S\))=Int \((R)_S\) for integer-valued polynomials, J. Pure Appl. Algebra, 125, 287-303 (1998) · Zbl 0897.13022 |
| [17] | Zafrullah, M., The \(v\)-operation and intersection of quotient rings of integral domains, Comm. Algebra, 13, 1699-1712 (1985) · Zbl 0573.13002 |
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