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Strong S-domains. (English) Zbl 0536.13001
An integral domain R is an S-domain if for each prime ideal P of R of height one the extension PR[X] to the polynomial ring in one indeterminate over R is also a prime ideal of height one. A commutative ring R is a strong S-ring if R/P is an S-domain for each prime ideal P of R. I. Kaplansky introduced the terminology of S-domains in his book ”Commutative rings” (2nd ed., 1974; Zbl 0296.13001). It is known that if a ring R in either Noetherian or Prüfer, then R is a strong S-domain and, moreover, the dimension formula holds, namely $$\dim R[X_ 1,X_ 2...X_ n]=n+\dim R$$, where dim R is the Krull dimension of R. Kaplansky observed that the dimension formula is a consequence of the strong S- property in the case that $$n=1$$. In general, it need not be the case that a polynomial ring extension of a strong S-ring is again a strong S-ring. Nevertheless, for certain special rings, the strong S-property is stable under polynomial ring extension (for example, Noetherian rings).
The main result of the paper is theorem 3.5 which states that $$D[X_ 1,X_ 2,...,X_ n]$$ is a strong S-domain if D is a Prüfer domain. Thus, we take the point of view that the dimension formula holds for Prüfer domains because of the stability of the strong S-property under polynomial ring extensions. - The proof of theorem 3.5 relies heavily on a theorem of M. Nagata [Theorem 2 in J. Math. Kyoto Univ. 5, 163- 169 (1966; Zbl 0163.034)]. The proof reduces to showing that $$Q_ 1[Y]\subset Q_ 2[Y]$$ are adjacent primes in V[X][Y] where $$Q_ 1\subset Q_ 2$$ are adjacent primes in V[X] where V is a valuation ring of Krull dimension one and where $$Q_ 1\neq(0)$$, $$Q_ 1\cap V=(0)$$, and $$Q_ 2\cap V=M$$, the maximal ideal of V. - An immediate corollary of Nagata’s theorem is the observation that if R is a Prüfer domain and Q is a prime ideal of $$R[X_ 1,..,X_ n]$$ of finite height, then the little rank of Q is equal to the height of Q. In particular, the authors observe in corollary 3.9 that saturated chains of prime ideals below Q have the same length, or in other words, if R is a Prüfer domain in which prime ideals have finite height, then $$R[X_ 1,...,X_ n]$$ is catenarian. Corollary 3.9 (or special cases of it) have subsequently been observed independently by several authors. Since we have relied on Nagata’s results so heavily some comments about them may be appropriate. The proof on page 165 of his theorem 1 may be correct as it stands, but a slight adjustment seems necessary. - Finally the statement in Nagata’s theorem 2 is easily extended to the case where $$A=R[a_ 1,a_ 2,...,a_ n]$$ is a finitely generated integral domain over a Prüfer domain R.

##### MSC:
 13B02 Extension theory of commutative rings 13G05 Integral domains 13B25 Polynomials over commutative rings 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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##### References:
 [1] Arnold, J.; Gilmer, R., The dimension sequences of a commutative ring, Amer. J. math., 96, 385-408, (1974) · Zbl 0302.13013 [2] Brewer, J., The ideal transform and overrings of integral domain, Math. Z., 107, 301-306, (1968) · Zbl 0167.03601 [3] Brewer, J.; Montgomery, P.R.; Rutter, E.A.; Heinzer, W.J., Krull dimensions of polynomial rings, (), No. 311 · Zbl 0249.13004 [4] Costa, D.; Mott, J.L.; Zafrullah, M., The D + XDS[X] construction, J. algebra, 53, 423-429, (1976) [5] Dobbs, D., Divided rings and going down, Pacific J. math., 67, 2, 353-363, (1976) · Zbl 0326.13002 [6] Fossum, R., The divisor class group of a Krull domain, (1973), Springer New York · Zbl 0256.13001 [7] Gilmer, R., Multiplicative ideal theory, (1972), Marcel Bekker New York · Zbl 0248.13001 [8] Gilmer, R., Two constructions of Prüfer domains, J. reine angew, math., 239, 153-162, (1970) · Zbl 0184.29102 [9] Gilmer, R.; Bastida, E., Overrings and divisorial ideals of rings of form D + M, Michigan math. J., 20, 79-95, (1973) · Zbl 0239.13001 [10] Harris, M., Some results on coherent rings, Proc. amer. math. soc., 18, 749-753, (1967) [11] Kaplansky, I., Commutative rings, (1974), Allyn and Bacon Boston, MA · Zbl 0203.34601 [12] Mott, J.L.; Schexnayder, M., Exact sequence of semivalue groups, J. reine angew. math., 283/284, 388-401, (1976) · Zbl 0347.13001 [13] Mott, J.L.; Zafrullah, M., On Prüfer υ-multiplication domains, Manuscripta math., 35, 1-26, (1981) · Zbl 0477.13007 [14] Nagata, M., Finitely generated rings over a valuation ring, J. math. Kyoto univ., 5, 2, 163-169, (1966) · Zbl 0163.03402 [15] Ratliff, I.J., Going between rings and contractions of saturated chains of primes, Rocky mountain J. math., 7, 4, 777-787, (1977) · Zbl 0372.13008 [16] Richman, F., Generalized quotient rings, Proc. amer. math. soc., 16, 794-799, (1965) · Zbl 0145.27406 [17] Seidenberg, A., On dimension theory of rings, Pacific J. math., 3, 513-522, (1953) · Zbl 0052.26902
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