About Euclidean rings

In this article all rings are commutative with unit, all modules are unitary. Given a ring A, its multiplicative group of units (i.e. invertible elements) is denoted by A*. The customary definition of a Euclidean ring is that it is a domain A together with a map F : A + N (the nonnegative integers) such that (1) I : p(a) for a, b E r3 (0); (2) given a, b E -‘-I, b m;’ 0, there exist q and Y in ‘4 such that a = bq + Y and dr) < v(b). The main interest of a Euclidean ring .4 is that it is principal: given any nonzero idea1 b in ,4. take a nonzero element b in b with the smallest possible value for p(b); then, for any n E b, (2) h s ows that a = bq $ r with Y = 0, whence b generates b. In this proof, (I) h as not been used, and any well-ordered set TV could replace N as the range of v. This has already been noticed by Th. lLIotzkin [7]. On the other hand the hypothesis that A is a domain does not seem to be essential. We therefore give the following definition: