On Pr\"ufer rings.

To find a necessary and sufficient condition for an integral domain to satisfy the following condition (C): (C) If A and B are torsion-free -modules, then A_B is also a torsion-free -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 denote an integral domain (with an identity). Instead of A_B, Tor₍^ (A, B), Hom_ (A, B), Ext_^n (A, B), we shall use simplified notations A B, Tor₍ (A, B), Hom (A, B), Ext^n (A, B), A and B being -modules. (See HA, for the definition of these functors).