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 ⊕ Rm≅Rn
for some m,n, is M 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 Hermlte ring is an elementary
divisor ring if and only if (A) has an affirmative answer for every module
M.
