Page 35 - Advanced Linear Algebra

Preliminaries 19

3 )(Distributivity ) For all Á Á  9 ,
² b ³ ~ b and ² b ³ ~ b

A ring is said to be commutative if ~ for all Á  9 . If a ring 9
9
contains an element with the property that

~ ~

for all  9 , we say that is a ring with identity . The identity is usually
9
denoted by .

A field is a commutative ring with identity in which each nonzero element
-
has a multiplicative inverse, that is, if  - is nonzero, then there is a  -
for which ~ .

Example 0.9 The set { ~ ¸ Á Á Ã Á c ¹ is a commutative ring under
addition and multiplication modulo
l ~ ² b ³ mod Á p ~ mod

The element  { is the identity.

Example 0.10 The set of even integers is a commutative ring under the usual
,
operations on , but it has no identity.
{
Example 0.11 The set C ²-³ is a noncommutative ring under matrix addition
and multiplication. The identity matrix is the identity for C ² - . ³
0
-
Example 0.12 Let be a field. The set - ´ % µ of all polynomials in a single
variable , with coefficients in - , is a commutative ring under the usual
%
operations of polynomial addition and multiplication. What is the identity for
-´%µ? Similarly, the set -´% Á Ã Á % µ of polynomials in variables is a

commutative ring under the usual addition and multiplication of polynomials.
Definition If 9 and : are rings, then a function ¢ 9 ¦ : is a ring
homomorphism if

² b ³ ~ b

² ³ ~ ² ³ ² ³
~
for all Á  9 .

Definition A subring of a ring is a subset of that is a ring in its own
:
9
9
right, using the same operations as defined on 9 and having the same
multiplicative identity as .
9