  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.
