Tensor products of modules and elementary equivalence (Q1077410)

In the first part of this paper the author investigates to what extent the elementary type of abelian groups A, B determine the elementary type of their tensor product (it is easy to see that tensorisation by an abelian group does not preserve elementary equivalence). The analysis is based on the following algebraic fact: let p be a prime and C, D be p- basic subgroups of A and B, respectively, then \(C\otimes D\) is a p-basic subgroup of \(A\otimes B\). A corollary is that if \(A\equiv A'\) and \(B\equiv B'\), then the reduced parts of \(A\equiv B\) and A'\(\equiv B'\) are elementarily equivalent. The second part of the paper deals with the question: what are the regular rings R such that tensorisation of R-modules preserves elementary equivalence. A counter-example is given with R boolean, but it is shown that if R is completely reducible then it is true. This is best understood in the light of \textit{G. Sabbagh}'s more recent result [Preprint 1984, to be published in the proceedings of the Conference in Memoriam Abraham Robinson, Yale, October 1984, ed. A. Macintyre] that, given any R, tensorisation by a pure-projective R-module preserves elementary equivalence, since, if R is artinian and semi-simple, which is the case here, then all R-modules are projective.