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

Abstract

In 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 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 M 0 ) are semi-inner. Therefore, for a wide range of rings, this solves Problem 12 left open in [22]; in particular, for Artinian (Noetherian, P I-) rings R, or a division semiring R, all automorphisms of Aut( R M 0 ) are semi-inner. April 30, 2005April 30, . 1991 Mathematics Subject Classification. Primary 16Y60, 16D90,16D99, 17B01; Secondary 08A35, 08C05. Date:Key words and phrases. free module, free semimodule over semiring, free modules over Lie algebras, semi-inner automorphism.

Automorphisms of categories of free algebras of some varieties

“…For example, the automorphisms are known for the varieties of all groups, all semigroups, all inverse semigroups, all Lie algebras, semimodules and modules (see [4,5,[9][10][11]). In each of these cases any automorphism of the category Θ 0 (V) turns out to be inner or close to inner.…”

Automorphisms of the endomorphism semigroup of a polynomial algebra

“…For every small category C, denote the group of all its automorphisms by Aut C. We distinguish the following classes of automorphisms of C. [8,15,20].) An automorphism ϕ : C → C is equinumerous if ϕ(D) ∼ = D for any object D ∈ Ob C; ϕ is stable if ϕ(D) = D for any object D ∈ Ob C; and ϕ is inner if ϕ and 1 C are naturally isomorphic, i.e., ϕ ∼ = 1 C .…”

“…Consider a constant morphism ν 0 : F (X) → F (X) such that ν 0 (x) = x 0 , x 0 ∈ F (X), for every x ∈ X . [8,13,16,20,23].) Let the free algebra F (X) generate a variety Θ, and ϕ ∈ St Aut Θ 0 .…”

“…Let Θ be a variety of algebras over a field K and F = F (X) be a free algebra from Θ generated by a finite subset X of some infinite universum X 0 . We refer to [17,18] (see also [8]) for the Universal Algebraic Geometry (UAG) notions used in our work.…”

“…The structure of the latter is determined by the group Aut End F . It should be mentioned that a problem of investigation of the groups Aut End F , F ∈ Θ, for different varieties Θ is quite interesting by itself and has been considered in many papers (see [1][2][3]5,[8][9][10][11][13][14][15][16][17][18][19]23]). …”