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 .
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.
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 .
A non empty subset S of a ring is a subring of R if S is a ring
- 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 .
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
- of is asubring.
- is this a subring of
- This is a commutative ring with identity
- Is a field?
- In , ?