# 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 Godel’s axiom of constructibility from two of Kaplansky test problems. For this, we introduce the construction of an (R,S)–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,
title = {{Kaplansky test problems for $R$-modules in ZFC}},
}