# Section 3.1 Part A: Rings

What methods are common among the following?

$\mathbb{Z}$, $k\mathbb{Z}$ where $k\ge 2$, $\mathbb{Z}_n$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$

###### Definition

A ring is a non empty set R with two operations, usually written as addition $a+b$, and multiplication $ab$, that satisfy the following axioms:

1. If $a,b\in R$, then $a+b\in R$ (Closure under Addition)
2. For all $a,b,c\in R$, we have $a+(b+c)=(a+b)+c$
3. For all $a,b\in R$ we have $a+b=b+a$ (Associative)
4. There exists an element $0_R\in R$ such that for all $a\in R$ we have $a+0_R=a=0_R+a$ (Additive Identity Element)
5. For each $a\in R$, the equation at $a+x=0_R$ has a solution $x\in R$
6. If $a,b\in R$, then $ab\in R$ (Closure under Multiplication)
7. For all $a,b\in R$, we have $a(bc)=(ab)c$ (Associativity in Multiplication)
8. For all $a,b,c\in R$, we have $a(b+c)=ab+ac$ and $(a+b)c=ac+bc$

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 ring R that satisfies

9. For all $a,b\in R$ we have $ab=ba$ (Multiplicative Identity)

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

1. $\mathbb{R}^3$ where it acts normal under addition but uses a cross product for multiplication:
1. 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. $M(\mathbb{R} )$ ( 2×2 real value matrices) where they act as usual under addition but $AB=B^T AB$
1. No. They are not associative. For example \begin{align*} (AB)C&= A (BC)\\ (B^T A B )C &= A (C^T BC)\\ C^T (B^T A B ) C & \neq (C^T B C )^T A (C^T B C)\end{align*}
3. $M(\mathbb{Z}$ acting as it normally does.
1. Yes.
4. $M(\mathbb{Q}$ acting as it normally does.
1. Yes.
5. $M(\mathbb{Z}_n$ acting as it normally does.
1. Yes.

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

###### Definition:

An integral domain is the commutative ring R with identity $1_R\neq 0_R$ ( In other words R is a ring with at least two elements) such that:

11.  For $a,b\in R$ if $ab=0_R$, then $a=0_R$ or $b=0_R$ (no zero divisor)

Here are a few examples that meet these conditions: $\mathbb{Z}_p$ (p is prime), $\mathbb{Z}$$\mathbb{Q}$$\mathbb{R}$$\mathbb{C}$.

###### Definition:

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

12.  For each $a\neq 0_R$ in R, the equation $ax=b$ has a solution x in R.

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