# Sh:421

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

We solve Kaplansky test problems, all three of which say that we do not have a decomposition theory. Let 0<m_1<m_2-1 and assume that R is not pure semisimple. As an application, we construct an R-module M such that: M^{n^1}\cong M^{n^2}\Longleftrightarrow \quad m_1<n^1\ \&\ m_1\leq n^2\ \&\ [(m_2-m_1)| (n^1-n^2)].Thereby giving a negative answer to the cube problem, which asks if M is isomorphic to M^3, does it follows that M is isomorphic to M^2? The innovation will be in generalizing from R: = \mathbb{Z} to not pure semisimple rings. Our results are in the ordinary set theory, ZFC. This drops Godelâ€™s axiom of constructibility from two of Kaplansky test problems. For this, we introduce the construction of an (R,S)â€“bimodule with few endomorphisms. This should be of particular interest. Among other things, we use a primary version of the black box. - Version 2022-06-07 (149p)

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} }