2005/05/09 by Yefim Katsov, Ruvim Lipyanski, Katsov, Yefim +4
Computer Science · Mathematics · #08A35 #16D90 #16D99 #16Y60 #17B01 #Category Theory (math.CT) #FOS: Mathematics #Geometric and Algebraic Topology #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.CT #math.RA #msc:08A35 #msc:16D90 #msc:16D99 #msc:16Y60 #msc:17B01 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0505151
25 pages
arxiv created 2005/05/09 · openalex publication_date 2005/05/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In algebraic geometry over a variety of universal algebras Θ, the group Aut(Θ0) of automorphisms of the category Θ0 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 RM of semimodules over an IBN-semiring R (an IBN-semiring is a semiring analog of a ring with IBN), all automorphisms of Aut(RM0) are semi-inner. Therefore, for a wide range of rings, this solves Problem 12 left open in \citeplotkin:slotuag; in particular, for Artinian (Noetherian, PI-) rings R, or a division semiring R, all automorphisms of Aut(RM0) are semi-inner.