Principle Ideal Domains and Various Ideals

In this section, we will see various kinds of ideals and their correspondence to the quotient ring. We spoil the upshot here:

Ideal	Quotient Ring
radical	reduced
prime	integral domain
maximal	field

Principal Ideal Domain

An integral domain is called a principal ideal domain if every ideal is principle.

e.g.

• Every field is a principal ideal domain. In fact, let be an ideal, if is non-trivial, then it contains a non-zero element . Since is a field, is a unit, and thus . So any field only has two trivial ideals: and ;
• is a PID, we will prove this in the next proposition;
• is a PID for any field ;
• for some field is not a principal ideal domain. The ideal is not principle;
• In , is not principle.
  

Proposition

is a principal ideal domain.

Proof Suppose is a non trivial ideal. There exists at least one positive integer in it. Pick the smallest positive integer . We claim . For any , by the remainder theorem, we have for some with . Since , it follows that . As is the smallest positive integer in , we must have . Therefore .

Remark

Indeed, any Euclidean domain is a principle ideal domain. (see proposition.)

Euclidean Algorithm

The gcd of polynomials , and the linear combination that gives it can be found by the following algorithm. Compute Then up to a unit is the gcd and backward substitution gives the linear combination.

Proposition

In an integral domain every prime is irreducible.

Proof Suppose is a prime with . Then implies or . Without loss of generality, suppose . By the equation, as well, so and are associates, thus is a unit.

Radical Ideals and Reduced Ring

Nilpotent & Idempotent

Let be a ring. is nilpotent if there is such that . is idempotent if .

Proposition

If is nilpotent then is invertible.

Prime and Maximal Ideals

Prime Ideal

Let be a ring. An ideal is prime if and implies that either or .

Theorem

Let be a commutative unital ring. is prime if and only if is an integral domain.

Proof is prime iff , implies , iff nonzero and has multiplication .

Maximal Ideal

Let be a ring. An ideal is maximal if and there is no other ideal such that .

e.g. is maximal iff for a prime . In general, we have the following theorem holds:

Theorem

Let be an ideal. Then is maximal iff is a field.

Proof is a field iff

Corollary

Every maximal ideal is prime. In a PID, conversely, every non-zero prime ideal is maximal.

Proof Every maximal ideal is prime is clear because every field is an integral domain.

Proposition

Let be a principle ideal domain, then is maximal if and only if is irreducible.

Proof In a PID, we have that iff . It follows that is maximal iff there is no proper divisor , which means is irreducible.

Corollary

If is a field, then is maximal iff is irreducible.

Proof is a PID, so is maximal iff is irreducible.