What methods are common among the following?

, where , , , $\mathbb{R}$,

Answering this question leads to the definition of a ring.

###### Definition

A ring is a non empty setRwith two operations, usually written as addition , and multiplication , that satisfy the following axioms:

If , then (Closure under Addition)For all , we have $a+(b+c)=(a+b)+c$For all we have (Associative)There exists an element such that for all we have (Additive Identity Element)For each , the equation at has a solutionIf , then (Closure under Multiplication)For all , we have (Associativity in Multiplication)For all , we have and

Note that commutativity isn’t explicitly written. That’s because multiplying on the left and right can be different. There some rings that are commutative. They are called *commutative rings.*

###### Definition:

A commutative ring is a ringRthat satisfies9.

For all we have (Multiplicative Identity)

All of these definitions may be a little confusing. But with a few examples will become easier to understand:

- where it acts normal under addition but uses a cross product for multiplication:
- No. It isn’t associative. For example $latex \begin{align*}\hat{i}\times (\hat{i}\times \hat{k})&=(\hat{i}\times \hat{i})\times \hat{k}\\ \hat{i}\times (-\hat{j} ) & = 0 \\ -\hat{k} & \neq 0\end{align*}#

- ( 2×2 real value matrices) where they act as usual under addition but
- No. They are not associative. For example

- acting as it normally does.
- Yes.

- acting as it normally does.
- Yes.

- acting as it normally does.
- Yes.

We didn’t explicitly mention commutitivity above, so it shall be defined here.

###### Definition:

An integral domain is the commutative ring

Rwith identity ( In other words R is a ring with at least two elements) such that:

11.For if , then or (no zero divisor)

Here are a few examples that meet these conditions: (p is prime), , , , .

###### Definition:

A field is a commutative ring with an identity that satisfies:

12. For each in

R, the equation has a solutionxinR.

So far only a few types of Rings have been mentioned. Below is a tree describing these various types: