2.5. Quotients 15

2.4. Ideals

A left ideal of an algebra A is a subspace I ⊆ A such that aI ⊆ I

for all a ∈ A. Similarly, a right ideal of an algebra A is a subspace

I ⊆ A such that Ia ⊆ I for all a ∈ A. A two-sided ideal is a

subspace that is both a left and a right ideal.

Left ideals are the same as subrepresentations of the regular rep-

resentation A. Right ideals are the same as subrepresentations of the

regular representation of the opposite algebra

Aop.

Below are some examples of ideals:

• If A is any algebra, 0 and A are two-sided ideals. An algebra

A is called simple if 0 and A are its only two-sided ideals.

• If φ : A → B is a homomorphism of algebras, then ker φ is

a two-sided ideal of A.

• If S is any subset of an algebra A, then the two-sided ideal

generated by S is denoted by S and is the span of ele-

ments of the form asb, where a, b ∈ A and s ∈ S. Similarly,

we can define S = span{as} and S

r

= span{sb}, the

left, respectively right, ideal generated by S.

Problem 2.4.1. A maximal ideal in a ring A is an ideal I = A

such that any strictly larger ideal coincides with A. (This definition

is made for left, right, or two-sided ideals.) Show that any unital

ring has a maximal left, right, and two-sided ideal. (Hint: Use Zorn’s

lemma.)

2.5. Quotients

Let A be an algebra and let I be a two-sided ideal in A. Then A/I

is the set of (additive) cosets of I. Let π : A → A/I be the quotient

map. We can define multiplication in A/I by π(a) · π(b) := π(ab).

This is well defined because if π(a) = π(a ), then

π(a b) = π(ab + (a − a)b) = π(ab) + π((a − a)b) = π(ab)

because (a − a)b ∈ Ib ⊆ I = ker π, as I is a right ideal; similarly, if

π(b) = π(b ), then

π(ab ) = π(ab + a(b − b)) = π(ab) + π(a(b − b)) = π(ab)