Strong S-domains.

*(English)*Zbl 0536.13001An 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.

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 |

##### Keywords:

extension of prime ideal to the polynomial ring; strong S-ring; dimension formula; Prüfer domain
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\textit{S. Malik} and \textit{J. L. Mott}, J. Pure Appl. Algebra 28, 249--264 (1983; Zbl 0536.13001)

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