A Ring can be seen as a generalization of a field where:

- multiplication is not necessarily commutative. If this is satisfied, we can call it a commutative ring.
- multiplication may not have inverse elements. If this is satisfied, we can call it a division ring.

Addition however has to be commutative and have inverses, i.e. it is an Abelian group.

The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set.

A polynomial ring is another example with the same properties as the integers.

The simplest non-commutative ring that is not a field is the set of all 2x2 matrices of real numbers:Note that $GL(n)$ is not a ring because you can by addition reach the zero matrix.

- we know that 2x2 matrix multiplication is non-commutative in general
- some 2x2 matrices have a multiplicative inverse, but others don't

Two ways to see it:

- a ring where inverses exist
- a field where multiplication is not necessarily commutative

$M_{n}(F_{q})$ accounts for them all, which we know how to do due to the classification of finite fields.

So we see that the classification is quite simple, much like the classification of finite fields, and in strict opposition to the classification of finite simple groups (not to mention the 2023 lack of classification for non simple finite groups!)

A ring where multiplication is commutative and there is always an inverse.

A field can be seen as an Abelian group that has two group operations defined on it: addition and multiplication.

And then, besides each of the two operations obeying the group axioms individually, and they are compatible between themselves accordin to the distributive property.

Basically the nicest, least restrictive, 2-operation type of algebra.

Examples:

One of the defining properties of algebraic structure with two operations such as ring and field:
This property shows how the two operations interact.

$a(b+c)=ab+ac$

A convenient notation for the elements of $GF(n)$ of prime order is to use integers, e.g. for $GF(7)$ we could write:
which makes it clear what is the additive inverse of each element, although sometimes a notation starting from 0 is also used:

$GR(7)={−3,−2,−1,0,1,2,3}$

$GR(7)={0,1,2,3,4,5,6}$

For fields of prime order, regular modular arithmetic works as the field operation.

For non-prime order, we see that modular arithmetic does not work because the divisors have no inverse. E.g. at order 6, 2 and 3 have no inverse, e.g. for 2:
we see that things wrap around perfecly, and 1 is never reached.

$0×2=01×2=22×2=43×2=04×2=25×2=4$

For non-prime prime power orders however, we can find a way, see finite field of non-prime order.

There's exactly one field per prime power, so all we need to specify a field is give its order, notated e.g. as $GF(n)$.

Every element of a finite field satisfies $x_{order}=x$.

It is interesting to compare this result philosophically with the classification of finite groups: fields are more constrained as they have to have two operations, and this leads to a much simpler classification!

As per classification of finite fields those must be of prime power order.

Video "Finite fields made easy by Randell Heyman (2015)" at youtu.be/z9bTzjy4SCg?t=159 shows how for order $9=3×3$. Basically, for order $p_{n}$, we take:For a worked out example, see: GF(4).

- each element is a polynomial in $GF(p)[x]$, $GF(p)[x]$, the polynomial ring over the finite field $GF(p)$ with degree smaller than $n$. We've just seen how to construct $GF(p)$ for prime $p$ above, so we're good there.
- addition works element-wise modulo on $GF(p)$
- multiplication is done modulo an irreducible polynomial of order $n$

Ciro Santilli tried to add this example to Wikipedia, but it was reverted, so here we are, see also: Section "Deletionism on Wikipedia".

This is a good first example of a field of a finite field of non-prime order, this one is a prime power order instead.

$4=2_{2}$, so one way to represent the elements of the field will be the to use the 4 polynomials of degree 1 over GF(2):

- 0X + 0
- 0X + 1
- 1X + 0
- 1X + 1

Note that we refer in this definition to anther field, but that is fine, because we only refer to fields of prime order such as GF(2), because we are dealing with prime powers only. And we have already defined fields of prime order easily previously with modular arithmetic.

Over GF(2), there is only one irreducible polynomial of degree 2:

$X_{2}+X+1$

Addition is defined element-wise with modular arithmetic modulo 2 as defined over GF(2), e.g.:

$(1X+0)+(1X+1)=(1+1)X+(0+1)=0X+1$

Multiplication is done modulo $X_{2}+X+1$, which ensures that the result is also of degree 1.

For example first we do a regular multiplication:

$(1X+0)×(1X+1)=(1×1)X_{2}+(1×1)X+(0×1)X+(0×1)=1X_{2}+1X+0$

Without modulo, that would not be one of the elements of the field anymore due to the $1X_{2}$!

So we take the modulo, we note that:
and by the definition of modulo:
which is the final result of the multiplication.

$1X_{2}+1X+0=1(X_{2}+X+1)+(0X+1)$

$(1X_{2}+1X+0)mod(X_{2}+X+1)=(0X+1)$

TODO show how taking a reducible polynomial for modulo fails. Presumably it is for a similar reason to why things fail for the prime case.

A function:

$R_{m}→R_{n}$

A vector field with a bilinear map into itself, which we can also call a "vector product".

Note that the vector product does not have to be neither associative nor commutative.

Examples: en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1035146107#Motivating_examples

- complex numbers, i.e. $R_{2}$ with complex number multiplication
- $R_{3}$ with the cross product
- quaternions, i.e. $R_{4}$ with the quaternion multiplication

An algebra over a field where division exists.

Notably, the octonions are not associative.