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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 toosimilar to the case $ R $ is a field.  We shall solve Kaplansky test problems, all three of which say that we do nothave decomposition theory, e.g., if the square of one module is isomorphic tothe cube of another it does not follows that they are isomorphic. Our resultsare in the ordinary et theory, ZFC. For this, we look at bimodules, i.e., thestructures which are simultaneously left $ R $-modules and right $ S $-moduleswith reasonable associativity, over a commutative ring $ T $ included in thecenters 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 theblack box.

Kaplansky Test Problems for $R$-Modules in ZFC