Automorphisms of categories of free modules, free semimodules, and free Lie modules

Y. Katsov, R. Lipyanski, B. Plotkin

Research output: Contribution to journalArticlepeer-review

18 Scopus citations

Abstract

In algebraic geometry over a variety of universal algebras , the group Aut(0) of automorphisms of the category 0 of finitely generated free algebras of is of great importance. In this article, 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 Plotkin (2002); in particular, for Artinian (Noetherian, PI-) rings R, or a division semiring R, all automorphisms of Aut(RM0) are semi-inner.

Original languageEnglish
Pages (from-to)931-952
Number of pages22
JournalCommunications in Algebra
Volume35
Issue number3
DOIs
StatePublished - 1 Mar 2007

Keywords

  • Free module
  • Free modules over Lie algebras
  • Free semimodule over semiring
  • Semi-inner automorphism

ASJC Scopus subject areas

  • Algebra and Number Theory

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