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Gilmer , Two cmistructions of Prüfer domams

Lemma 3. 5, there is a nonzero finitely generated ideal of / such that А1^= Л^ + М^ for each i. Hence [л{А) = oc and ^ is onto. If P is the subgroup of F consisting of zero principal fractional ideals of /, then it is clear that P g ker //. However, \i F ^ ker /^, then choose a nonzero element d oi J such that dF — 5 is an integral ideal of /. Then BJ^ = d • FJ^ is principal in J^ for each i. By Theorem 3. 5 b), it follows that Б, and hence F, is principal. Thus ker /л = P and F/P, the class group of /, is isomorphic to G as asserted.

We remark that in case D^ is Prüfer with quotient field ÜC, D^ and J^ have the same class group ([7], p. 562).

Acknowledgement . The author's investigation of the construction in § 2 was motivated by a remark of W. Heinzer, who reported to the author that A. Dress, during a visit to Louisiana State University, had stated that if Z is a field in which — 1 is not a square, then the subring of К generated by {1/(1 + t^)}t^K i^ ^ Prüfer domain whose set of valuation overrings is the set of valuation rings on К for which — 1 is not a square in the residue field.

References

[ 1 ] H. S. Butts and W. W. Smith, Prüfer rings, Math. Zeit. 95 (1967), 196—211.

[ 2 ] Я. S. Butts and W. W. Smith, On the integral closure of a domain, J. Sei. Hiroshima Univ. Ser. A—I, 30

( 1966 ) , 117—122. [3] E. D. Davis, Overrings of commutative rings. II. Integrally closed overrings, Trans. Amer. Math. Soc. 110

( 1964 ) , 196—212. [4] P. Eakin, The converse to a well-known theorem on Noetherian rings. Math. Ann. 177 (1968), 278—282. [5] D. Estes and J. Ohm, Stable range in commutative rings, J. Algebra 7 (1967), 343—362. [6] R. Gilmer, Overrings of Prüfer domains, J. Algebra 4 (1966), 331—340. [7] R. Gilmer, Multiplicative ideal theory, Queen's Papers on Pure and AppHed Mathematics, No. 12, Kingston,

Ontario , Canada 1968. [8] R. Gilmer, On a condition of J. Ohm's for integral domains, Canad. J. Math. 20 (1968), 970—983. [9] R. Gilmer and W. Heinzer, Overrings of Prüfer domains. II, J. Algebra 7 (1967), 281—302. [10] R. Gilmer and W. Heinzer, On the complete integral closure of a domain, J. Austral. Math. Soc. 6 (1966),

351—361 .

[ 11 ] R. Gilmer and W. Heinzer, On the number of generators of an invertible ideal, to appear in J. Algebra. [12] R. Gilmer and W. Heinzer, Primary ideals and valuation ideals. II, Trans. Amer. Math. Soc. 131 (1968), 149—162.

[ 13 ] R. Gilmer and J. Ohm, Integral domains with quotient overrings. Math. Ann. 153 (1964), 97—103. [14] 0. Helmer, Divisibility properties of integral functions, Duke Math. J. 6 (1940), 345—356. [15] C. U. Jensen, On characterizations of Prüfer rings, Math. Scand. 13 (1963), 90—98. [16] W, Krull, Beiträge zur Arithmetik kommutativer Integritätsbereiche, Math. Z. 41 (1936), 545—577. [17] M. Nagata, On the theory of Henselian rings, Nagoya Math. J. 5 (1953), 45—57. [18] M. Nagata, Local rings. New York 1962.

[ 19 ] J. Ohm, Primary ideals in Prüfer domains, Canad. J. Math. 18 (1966), 1024—1030.

[ 20 ] H. Prüfer, Untersuchungen über Teilbarkeitseigenschaften in Körpern, J. Reine Angew. Math. 168 (1932), 1-36.

Florida State University, Department of Mathematics, Tallahassee, FL 32306, USA

Eingegangen 11. Februar 1969