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

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
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