Page 485 - Advanced Linear Algebra

An Introduction to Algebras 469

Now, if < £ ¸ ¹ , then there is a "  < for which " ~ c and
" ~ c "
" ~ c "
" ~ c "

The third equation is "² ³ ~ c² ³" and so
"² ³ ~ c² ³" ~ " ~ c"

whence " ~ , which is false. Hence, < ~ ¸ ¹ and
s
s
s
+ ~ s l l l ~ i
This completes the proof.
Exercises
1. Prove that the subalgebra generated by a nonempty subset of an algebra
?
( is the subspace spanned by the products of finite subsets of elements of
?:

º?» alg ~ º% Ä% “ %  ?»

2. Verify that the group algebra -´.µ is indeed an associative algebra over .
-
3. Show that the kernel of an algebra homomorphism is an ideal.
4. Let be a finite-dimensional algebra over and let be a subalgebra.
-
(
)
Show that if  ) is invertible, then c  ) .
(
5. If is an algebra and ‹ : ( is nonempty, define the centralizer * ( ² : ³ of
: ( to be the set of elements of that commute with all elements of . Prove
:
(
that *²:³ is a subalgebra of .
(
is not an algebra over any field.
6. Show that {
7. Let (~ -´ µ be the algebra generated over by a single algebraic element
-
. Show that ( is isomorphic to the quotient algebra -´%µ°º ²%³», where
º ²%³» is the ideal generated by ²%³  -´%µ. What can you say about
²%³? What is the dimension of (? What happens if is not algebraic?
8. Let . ~ ¸ ~ ÁÃÁ ¹ be a finite group. For %  -´.µ of the form

% ~ bÄb

let ;²%³ ~ bÄb . Prove that ;¢ -´.µ ¦ - is an algebra
homomorphism, where is an algebra over itself.
-
9. Prove the first isomorphism theorem of algebras: A homomorphism

--algebras induces an isomorphism ¢( ¦ ) of ¢(°ker ² ³ š im ² ³
defined by ² ker ² ³³ ~ .
10. Prove that the quaternion field is an -algebra and a field. Hint : For
-
% ~ b b b £