Re: principal ideals

See

Wiegand, Roger; Wiegand, Sylvia
Commutative rings whose finitely generated modules are direct sums of
cyclics. Abelian group theory (Proc. Second New Mexico State Univ.
Conf., Las Cruces, N.M., 1976), pp. 406--423. Lecture Notes in Math.,
Vol. 616, Springer, Berlin, 1977.

From the MathSciNet review:

The question of which commutative rings have the property that every
finitely generated module is a direct sum of cyclic modules has been
around for many years. We will call these rings FGC rings. The problem
originated in I. Kaplansky's papers [Trans. Amer. Math. Soc. 66
(1949), 464--491; MR0031470 (11,155b); ibid. 72 (1952), 327--340;
MR0046349 (13,719e)], in which it was shown that a local domain is FGC
if and only if it is an almost maximal valuation ring. The final
solution of the problem came in the work of W. Brandal and the first
author in 1976. The paper under review is an excellent exposition of
that result, with complete details.

An integral domain is said to be $h$-local if every nonzero ideal is
contained in only finitely many maximal ideals and every nonzero prime
ideal is contained in a unique maximal ideal. This notion was
introduced by E. Matlis and used by him [ibid. 125 (1966), 147--179;
MR0201465 (34 \#1349)] to show that an $h$-local domain is FGC if and
only if it is Bézout (i.e., finitely generated ideals are principal)
and its localization at each maximal ideal is an almost maximal
valuation ring. Ten years later, it was finally shown by W. Brandal
that these are the only FGC domains. Within a few months, the first
author had completed the description of all FGC commutative rings. The
result is that a commutative ring is an FGC ring if and only if it is
a finite product of indecomposable FGC rings and $R$ is an
indecomposable FGC ring if and only if it has a unique minimal prime
ideal $P$ such that (a) $R/P$ is a Bézout domain whose localizations
are almost maximal valuation rings, and (b) the ideals contained in $P
$ are linearly ordered by inclusion.


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