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, semiinner 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 semiinner. We thus prove that for a variety $_{R}\mathcal{M}$ of semimodules over an IBNsemiring $R$ (an IBNsemiring is a semiring analog of a ring with IBN), all automorphisms of $Aut(_{R}\mathcal{M}^{0})$ are semiinner. 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 semiinner.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2005
 DOI:
 10.48550/arXiv.math/0505151
 arXiv:
 arXiv:math/0505151
 Bibcode:
 2005math......5151K
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Category Theory;
 16Y60;
 16D90;
 16D99;
 17B01;
 08A35
 EPrint:
 25 pages