# Automorphisms of Categories of Free Modules, Free Semimodules, and Free Lie Modules∗ Academic Article

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### 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.

### publication date

• January 1, 2007