## 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 language | English |
---|---|

Pages (from-to) | 931-952 |

Number of pages | 22 |

Journal | Communications in Algebra |

Volume | 35 |

Issue number | 3 |

DOIs | |

State | Published - 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