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 , ?