Page 472 - Advanced Linear Algebra
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456 Advanced Linear Algebra
The usual terms monomorphism, epimorphism, isomorphism, embedding,
endomorphism and automorphism apply to algebras with the analogous meaning
as for vector spaces and modules.
Example 18.6 Let be an -dimensional vector space over . Fix an ordered
=
-
8
=
basis for . Consider the map ¢ B ² = ³ ¦ C ² - ³ defined by
²³ ~ ´ µ 8
8
is the matrix representation of with respect to the ordered basis .
where ´µ 8
This map is a vector space isomorphism and since
´ µ 8 ~ ´ µ 8 ´ µ 8
it is also an algebra isomorphism.
Another View of Algebras
If is an algebra over , then contains a copy of . Specifically, we define a
-
(
-
(
function ¢- ¦ ( by
~
for all - , where ( is the multiplicative identity. The elements are in
(
the center of , since for any ( ,
² ³ ~ ² ³ ~
and
² ³ ~ ² ³ ~
Thus, . To see that is a ring homomorphism, we have
¢- ¦ A²(³
² ³ ~ h ~
-
-
² b ³ ~ ² b ³ ~ b ~ ² ³ b ² ³
² ³ ~ ² ³ ~ ² ³ ~ ² ³ ~ ² h ² ³³ ~ ² h ³ ² ³ ~ ² ³ ² ³
Moreover, if ~ and £ , then
c
~ ² ³~ h ~
-
and so provided that £ in , we have ~ . Thus, is an embedding.
(
Theorem 18.1
)
(
1 If is an associative algebra over and if - £ in , then the map
(
¢- ¦ A²(³ defined by
~
-
is an embedding of the field into the center ² A ( ³ of the ring . Thus, -
(
can be embedded as a subring of A²(³ .
