On a problem of Kaplansky (Q5903264)

A well-known result of abelian group theory due to \textit{B. Jónsson} [Math. Scandinav. 5, 230-235 (1958; Zbl 0081.262)] states that torsion- free groups of finite or countable rank fail to satisfy the Krull-Schmidt property. The author here provides several new examples, constructed in a manner similar to Jónsson's, of groups which fail \textit{I. Kaplansky}'s Test Problems [Infinite abelian groups (1969; Zbl 0194.044)]. He also extends a theorem of Kaplansky on finite-rank modules over a discrete valuation ring.