Page 38 - Advanced Linear Algebra
P. 38
22 Advanced Linear Algebra
To see what the equivalence classes look like, observe that
´ µ ~ ¸ 9 ¹
~¸ 9 c :¹
~¸ 9 ~ b for some :¹
~¸ b :¹
~ b :
The set
b : ~¸ b :¹
is called a coset of in . The element is called a coset representative for
:
9
b:.
:
Thus, the equivalence classes for congruence mod are the cosets b : of :
in . The set of all cosets is denoted by
9
9°: ~ ¸ b : 9¹
:
This is read “ mod .” We would like to place a ring structure on 9 ° . :
9
Indeed, if is a subgroup of the abelian group , then ° 9 9 : is easily seen to be
:
an abelian group as well under coset addition defined by
² b:³b² b :³ ~ ² b ³b:
In order for the product
² b:³² b:³ ~ b:
to be well-defined, we must have
Z
Z
b :~ b :¬ b :~ b :
or, equivalently,
Z
Z
c : ¬ ² c ³ :
:
But c Z may be any element of and may be any element of and so this
9
condition implies that must be an ideal. Conversely, if is an ideal, then
:
:
coset multiplication is well defined.
Theorem 0.21 Let be a commutative ring with identity. Then the quotient
9
9° is a ring under coset addition and multiplication if and only if is an
?
?
ideal of . In this case, ° 9 ? is called the quotient ring of modulo , where
9
?
9
addition and multiplication are defined by
² b:³b² b :³ ~ ² b ³b:
² b:³² b:³ ~ b:
