Documentation

Mathlib.Algebra.Ring.Aut

Ring automorphisms #

This file defines the automorphism group structure on RingAut R := RingEquiv R R.

Implementation notes #

The definition of multiplication in the automorphism group agrees with function composition, multiplication in Equiv.Perm, and multiplication in CategoryTheory.End, but not with CategoryTheory.comp.

Tags #

ring aut

@[reducible, inline]
abbrev RingAut (R : Type u_1) [Mul R] [Add R] :
Type u_1

The group of ring automorphisms.

Equations
instance RingAut.instGroup (R : Type u_1) [Mul R] [Add R] :

The group operation on automorphisms of a ring is defined by fun g h => RingEquiv.trans h g. This means that multiplication agrees with composition, (g*h)(x) = g (h x).

Equations
  • One or more equations did not get rendered due to their size.
instance RingAut.instInhabited (R : Type u_1) [Mul R] [Add R] :
Equations
def RingAut.toAddAut (R : Type u_1) [Mul R] [Add R] :

Monoid homomorphism from ring automorphisms to additive automorphisms.

Equations
def RingAut.toMulAut (R : Type u_1) [Mul R] [Add R] :

Monoid homomorphism from ring automorphisms to multiplicative automorphisms.

Equations
def RingAut.toPerm (R : Type u_1) [Mul R] [Add R] :

Monoid homomorphism from ring automorphisms to permutations.

Equations