To find a necessary and sufficient condition for an integral domain Λ to satisfy the following condition (C) : (C) If A and B are torsion-free Λ -modules, then A⊗ΛB is also a torsion-free Λ -module. This is a problem recently proposed by M. Nagata.1) We know, following J. Dieudonn\'e,2) that (C) is satisfied by any Dedekind ring, and more generally by any Pr\"ufer ring, as is shown by H. Cartan and S. Eilenberg in their recent publication.3) In this paper, we shall prove conversely that a ring satisfying (C) is neces- sarily a Pr\"ufer ring (Theorem 2). This will solve the above pro- blem completely, and at the same time yield a characterization of Pr\"ufer rings.4) Let Λ denote an integral domain (with an identity). Instead of A⊗ΛB, TorₙΛ(A, B), HomΛ(A, B), ExtΛⁿ(A, B) , we shall use simplified notations A⊗ B, Torₙ(A, B), $Hom(A, B),$ Extⁿ(A, B) , A and B being Λ -modules. ( See $HA$, for the definition of these functors).
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Akira Hattori (1957) studied this question.