Page 471 - Advanced Linear Algebra
P. 471
An Introduction to Algebras 455
Ideals and Quotients
In defining the notion of an ideal of an algebra , we must consider the fact that
(
( may be noncommutative.
Definition A ( two-sided ideal of an associative algebra ( is a nonempty
)
subset of that is closed under addition and subtraction, that is,
0
(
Á 0 ¬ b Á c 0
and also left and right multiplication by elements of , that is,
(
0Á Á ( ¬ 0
The ideal generated by a nonempty subset ? of ( is the smallest ideal
containing and is equal to
?
º?» ideal ~ H % % ?Á Á ( I
~
Definition An algebra is simple if
(
)
(
1 The product in is not trivial, that is, £ for at least one pair of
elements Á (
)
2 ( has no proper nonzero ideals.
Definition If is an ideal in , then the quotient algebra is the quotient
(
0
ring/quotient space
(°0 ~ ¸ b 0 (¹
with operations
² b0³b² b 0³ ~ ² b ³b0
² b0³² b0³ ~ b0
² b0³ ~ b0
-
where Á ( and - . These operations make (°0 an -algebra.
Homomorphisms
Definition If ( and ) are - -algebras a map ¢ ( ¦ ) is an algebra
homomorphism if it is a ring homomorphism as well as a linear
transformation, that is,
Z
Z ² b ³ ~ Z b Á Z ² ³ ~ ² ³² ³Á ~
and
² ³ ~ ² ³
for - .
