# Sh:421

- Asgharzadeh, M., Golshani, M., & Shelah, S.
*Kaplansky test problems for R-modules in ZFC*. Preprint. arXiv: 2106.13068 -
Abstract:

We conclude a long-standing research program in progress since 1954 by giving negative answers to test problems Kaplansky. Among these problems, the first was largely open, but the others were known to be consequences of Jensenâ€™s diamond principle and therefore impossible to answer affirmatively. Let R be a left non-pure semisimple, and let m> 1 be a natural number. For example, we construct an R-module M such that M^n\cong M if and only if m divides n-1, thus solving the first test problem in the negative. As an application, we also construct an R-module M of arbitrary size such that M^{n_1}\cong M^{n_2} if and only if m divides (n_1-n_2), giving a strongly negative answer to the cube problem of whether an R-module M which is isomorphic to M^3 must be isomorphic to its square M^2? We will treat the other two problems similarly. The crux of our method is to construct a ring S and an (R, S)-bimodule with few endomorphisms, for which we rely heavily on techniques from algebra and set theory, in particular the black box. - Version 2022-06-07 (153p)

Bib entry

@article{Sh:421, author = {Asgharzadeh, Mohsen and Golshani, Mohammad and Shelah, Saharon}, title = {{Kaplansky test problems for $R$-modules in ZFC}}, note = {\href{https://arxiv.org/abs/2106.13068}{arXiv: 2106.13068}}, arxiv_number = {2106.13068} }