Anybody who has done a course in abstract algebra should be familiar with rings and fields. Let’s start with the definitions.

Aringis a setRtogether with binary operations + (addition) and × (multiplication) such thatSome authors also insist that a ring

Ris an Abelian group under + with identity element 0 (called zero),- × is associative, and is distributive over + (both from the left and from the right)
Rmust contain a multiplicative identity 1 (called unity).

Afieldis a setFwith similar addition and multiplication operations such thatThus all fields are rings: {rings} ⊇ {fields}.

Fis a ring (with unity) under + and ×,- F\{0} forms an Abelian group under ×.

However the chain is much more interesting than that.

Note that ring multiplication need not be commutative. If it is, we say that the ring is acommutative ring. Matrices (i.e. square matrices) provide the best examples of noncommutative rings, while the integers furnish the most familiar example of a commutative ring.

Moreover, with matrices, it is possible for the product of two nonzero matrices to be zero. This is not the case with integers: ifmandnare integers, andmn= 0, then eitherm= 0 orn= 0.

Anintegral domainis a commutative ringR(with unity) such that .

Another property of the integers is the unique factorization: every integern> 1 can be written uniquely as a product of primes. This idea can be extended to integral domains. First, we need some definitions. (We’ll also drop the × symbol for ring multiplication and just writeabfora×b.)

IfRis a commutative ring (with unity) and , we say thatadividesbiff there exists such that . A unit (not to be confused with “unity”ť) inRis an element that divides 1. An element is said to be an irreducible iffris not 0 or a unit and for any eitheraorbis a unit.

Now suppose given we can write where the ’s are irreducibles; suppose moreover that if , where the ’s are irreducibles, thenm=nand there is a permutation of {1,…,m} such that for some unit . Then we say thatais uniquely factorized inR. Aunique-factorization domain(orUFD) is an integral domain in which every element that is not 0 or a unit can be uniquely factorized.

Hence is a UFD. The units of are ±1, and the irreducibles are ±p, wherepis a prime number. Actually, in ring theory, primes are defined differently from the way integer prime numbers are defined. IfRis a commutative ring with unity, an element is a prime iffpis not 0 or a unit, and for any wheneverpdividesab, it follows thatpdividesaorpdividesb. It turns out that every prime is an irreducible in an integral domain; in a UFD, every irreducible is also a prime.

IfRis a commutative ring, an ideal inRis a subsetIofRsuch thatIis an additive subgroup ofRand . If there is an element such that , thenIis said to be a principal ideal (generated by ). An integral domain in which every ideal is principal is called aprincipal-ideal domain(orPID).

is a PID. Given any integern, the set is an ideal; conversely every ideal in is of this form. As a matter of fact,every PID is a UFD.

Yet another familiar property of the integers is the Euclidean property: ifmandnare integers andn≠ 0, then there exist integersqandrsuch thatm=qn+rand 0 ≤r< |n|.

In general, given a ringR, suppose there exists a function (called a norm function); if such that and eitherr= 0 or , thenRis said to satisfy the Euclidean property. (For the integers, the norm function is the absolute-value function.) AEuclidean domainis an integral domain that satisfies the Euclidean property.

And indeed,every Euclidean domain is a PID.

Finally, note that every fieldFis a Euclidean domain. , we have (where is the multiplicative inverse ofb) – and any function can be the norm function.

So, we have a much more interesting chain as follows: {rings} ⊇ {commutative rings (with unity)} ⊇ {integral domains} ⊇ {UFDs} ⊇ {PIDs} ⊇ {Euclidean domains} ⊇ {fields} :-D