The next part of rings deals with an idea called a Cartesian Product. A simple example would be the ordered pairs used to make two dimensional plots: (x,y). This ordered pair is the cartesian product .
Definition:
The Cartesian Product of two non empty sets R and S is the set of ordered pairs
For Example: If then
In a cartesian product addition and multiplication are defined
Note that the cartesian product of two commutative rings has a product that is also commutative, and the product of two fields is not necessarily a field.
Theorem 3.1:
Let R and S be rings. The Cartesian product $$R\times S$$ is also a ring with
If both R and S have identities
then
has an identity
.
Definition:
A non empty subset S of a ring is a subring of R if S is a ring
Example:
is a subring of
- The odd integers (not closed under addition and does not contain the zero element)
is a subring of
which is a subring of
which is a subring of
.
Theorem 3.2:
Suppose R is a ring. A subset S of R is a subring of R if
-
S is closed under addition
-
S is closed under multiplication
-
-
For each
, the equation
has a solution x in S
Example:
of
is asubring.
is this a subring of
implies
- This is a commutative ring with identity
- Is
a field?
- In
,
?