Kaplansky’s problem on valuation rings. (English) Zbl 0691.13027

Fuchs, Laszlo; Shelah, Saharon

Proc. Am. Math. Soc. 105, No. 1, 25-30 (1989).

The authors prove that, for a valuation domain R and an R-module U, the ring \(S=R\otimes U\) with multiplication defined via \((r,u)(r',v)=(rr',r'u+rv)\) is a homomorphic image of a valuation domain if and only if U is a standard uniserial divisible R-module. Then they establish the existence of valuation domains (within ZFC) for which non- standard uniserial divisible modules exist. The proof is via multisorted models and forcing arguments and uses earlier results by the authors (some of them involving the diamond principle). There is still a need for purely algebraic proofs of these results as well as concrete constructions of the specific rings and modules within the framework of ZFC.

Reviewer: R.M.Dimitrić

MSC:

13L05	Applications of logic to commutative algebra
13F99	Arithmetic rings and other special commutative rings
13A05	Divisibility and factorizations in commutative rings
03H15	Nonstandard models of arithmetic

Keywords:

Kaplansky’s problem; valuation domain; non-standard uniserial divisible modules; diamond principle; ZFC