A domain R is said to be FGC
if every finitely generated R-module decomposes into a direct sum of cyclic
submodules. The main result is: if R is a domain with quotient field Q, then R is
FGC if and only if all of the following three conditions are satisfied: (1) R is Bezout,
(2) Q∕R is an injective R-module, and (3) there does not exist a continuous
embedding of βN into spec R relative to the patch topology of spec R. This result
is also true if (3) is replaced: (3′) every nonzero element of R is an element of only
finitely many maximal ideals of R. Using entire functions, there exists an example of
a domain satisfying (1) and (2), but not satisfying (3). Also presented are
some partial results towards generalizing the main result to commutative
rings.
