Automorphisms of Categories of Free Modules, Free Semimodules, and Free Lie Modules

Abstract

In algebraic geometry over a variety of universal algebras $\Theta$, the group $Aut({\Theta }^0)$ of automorphisms of the category ${\Theta }^0$ of finitely generated free algebras of $\Theta$ is of great importance. In this paper, semi-inner automorphisms are defined for the categories of free (semi)modules and free Lie modules; then, under natural conditions on a (semi)ring, it is shown that all automorphisms of those categories are semi-inner. We thus prove that for a variety ${}_R\mathcal{M}$ of semimodules over an IBN-semiring $R$ (an IBN-semiring is a semiring analog of a ring with IBN), all automorphisms of $Aut{(}_R{\mathcal{M}}^0)$ are semi-inner. Therefore, for a wide range of rings, this solves Problem 12 left open in \cite{plotkin:slotuag}; in particular, for Artinian (Noetherian, $PI$-) rings $R$, or a division semiring $R$, all automorphisms of $Aut{(}_R{\mathcal{M}}^0)$ are semi-inner.

25 pages

References in corpus (2)

• Automorphisms of categories of free modules and free Lie algebras
• A generalization and a new proof of Plotkin's reduction theorem