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)
Previous Page Next Page