Abstract Algebra: Theory and Applications · Problem 35

Polynomials — Thomas Judson

Mathematics — Abstract Algebra · inner-automorphisms-rings

Problem

Let

R

be a ring with identity. (a) Let

u

be a unit in

R

. Define a map

i_u : R \to R

r \mapsto uru^{-1}

. Prove that

i_u

is an automorphism of

R

. Such an automorphism of

R

is called an inner automorphism of

R

. Denote the set of all inner automorphisms of

R

\text{Inn}(R)

. (b) Denote the set of all automorphisms of

R

\text{Aut}(R)

. Prove that

\text{Inn}(R)

is a normal subgroup of

\text{Aut}(R)

. (c) Let

U(R)

be the group of units in

R

. Prove that the map

\phi : U(R) \to \text{Inn}(R)

defined by

u \mapsto i_u

is a homomorphism. Determine the kernel of

\phi

. (d) Compute

\text{Aut}(\mathbb{Z})

\text{Inn}(\mathbb{Z})

, and

U(\mathbb{Z})

Intuitive explanation

Inner automorphisms are ring isomorphisms built by conjugating elements with units—they're the "symmetries you get for free" from the ring's own invertible elements. The key insight is that conjugation by a unit preserves all ring structure (addition and multiplication), and the set of all such conjugations forms a normal subgroup of all automorphisms because any automorphism can "conjugate away" an inner automorphism. Part (c) asks you to see that different units might give the same inner automorphism, so the kernel captures exactly which units act trivially by conjugation—those that commute with everything.

Intuitive explanation

Related theory

Inner Products And Inner Product Spaces

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