Researches

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.

FC-Pure Injectivity and FC-Pure Flatness and Its Relationship to Two Famous Problems of Kothe and Kaplansky

Let S be a class of left R-modules.  An R-module M is said to be S-pure injective (resp. S-pure projective) if M has the injective (resp. projective) property respect to each S-pure exact sequence. Also, an R-module M is said to be S-pure flat if M has the flat property respect to each S-pure exact sequence. As  Puninski (1994), let us fix a few of the most common purities S of R-modules.