Finitely generated modules over Bezout rings

Let R be a Bezout ring (a commutative ring in which all finitely generated ideals are principal), and let M be a finitely generated R -module. We will study questions of the following sort: (A) If every localization of M can be generated by n elements, can M itself be generated by n elements? (B) If M 0 R m = R n for some m, n, is Af necessarily free? (C) If every localization of M has an element with zero annihilator, does M itself have such an element? We will answer these and related questions for various familiar classes of Bezout rings. For example, the answer to (B) is "no" for general Bezout rings but "yes" for Hermite rings (defined below). Also, a Hermite ring is an elementary divisor ring if and only if (A) has an affirmative answer for every module M.