Model-theoretic $K_1$ for modules over semisimple rings: (weak) Morita invariance

Abstract

This paper is a sequel to a paper by the same authors, where they defined $K$-groups of model-theoretic structures, and computed $K_1$ of free modules over PIDs. In this paper, we compute $K_1$ of a right $M_q(R)$-module $M$, where $R$ is a division ring, $q\geq1$, and $|M_q(R)|\neq 2$. As a consequence, we obtain a (weak) Morita invariance $K_1(R_R)\cong K_1((M_q(R))_{M_q(R)})$ for all division rings $R$ and $q\geq 1$. Finally, we compute $K_1$ of a module over a semisimple ring by showing that the model-theoretic $K_1$ commutes with finite product of modules. We also show that the algebraic $K_1$ of a finite product of infinite matrix rings embeds into the model-theoretic $K_1$ of their right regular modules.