Kaplansky Test Problems for $R$-Modules in ZFC

TL;DR

This paper addresses Kaplansky test problems for R-modules within ZFC, demonstrating that certain decomposition properties do not hold, by analyzing bimodules and their endomorphism rings.

Contribution

It provides solutions to Kaplansky test problems for R-modules in ZFC using bimodule analysis and constructs specific bimodules with minimal endomorphism rings.

Findings

Decomposition theory does not hold for certain R-modules.

Constructed bimodules with minimal endomorphism rings.

Solved all three Kaplansky test problems in ZFC.

Abstract

Fix a ring $ R $ and look at the class of left $ R $-modules and naturally, we restrict ourselves to the case of rings such that this class is not too similar to the case $ R $ is a field. We shall solve Kaplansky test problems, all three of which say that we do not have decomposition theory, e.g., if the square of one module is isomorphic to the cube of another it does not follows that they are isomorphic. Our results are in the ordinary et theory, ZFC. For this, we look at bimodules, i.e., the structures which are simultaneously left $ R $-modules and right $ S $-modules with reasonable associativity, over a commutative ring $ T $ included in the centers of $ R $ and of $ S $. Eventually, we shall choose $ S $ to help solve each of the test questions. But first, we analyze what can be the smallest endomorphism ring of an $(R, S)$-bimodule. We construct such bimodules by using a simple version of the black box.