Kaplansky Test Problems for $R$-Modules 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 set 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.