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

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.