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

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dc.creatorKatsov, Yefim
dc.creatorLipyanski, Ruvim
dc.creatorPlotkin, Boris
dc.date2005-05-09
dc.date.accessioned2026-07-07T05:19:43Z
dc.date.available2026-07-07T05:19:43Z
dc.descriptionIn algebraic geometry over a variety of universal algebras $Θ$, the group $Aut(Θ^{0})$ of automorphisms of the category $Θ^{0}$ of finitely generated free algebras of $Θ$ 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.
dc.description25 pages
dc.identifierhttps://arxiv.org/abs/math/0505151
dc.identifierhttp://arxiv.org/abs/math/0505151
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/75121
dc.subjectRings and Algebras
dc.subjectCategory Theory
dc.subject16Y60; 16D90; 16D99; 17B01; 08A35
dc.titleAutomorphisms of Categories of Free Modules, Free Semimodules, and Free Lie Modules
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