Commutative Rings Whose Finitely Generated Ideals Decompose into Direct Sums of Cyclic Ideals
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. Kaplanskys papers [5], [6];in which it was shown that a local domain is FGC if and only if it is an almost maximal valuation ring. For several years, this is one of the major open problems in the theory. The problem studied and solved in 1970s by various authors (R. S. Pierce[7], Brandal [2], Shores-R. Wiegand [8], S. Wiegand [10], Brandal-R.