Principle Ideal Domains

Principal Ideal Domain

A integral domain $R$ is called a principal ideal domain if every ideal is principal.

For example, both $\mathbb{Z}$ and the ring of polynomials $F[x]$ over a field $F$ are PIDs:

Proposition

$\mathbb{Z}$ is a principal ideal domain.

Proof Suppose $I◃\mathbb{Z}$ is a non trivial ideal. There exists at least one positive integer in it. Pick the smallest positive integer $n\in I$. We claim $I=(n)$. For any $m\in I$, by the remainder theorem, we have $m=qn+r$ for some $q,r\in \mathbb{Z}$ with $0\le r<n$. Since $n,m\in I$, it follows that $r=m-qn\in I$. As $n$ is the smallest positive integer in $I$, we must have $r=0$. Therefore $m=qn\in (n)$. $\square$

Remark

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

Proposition

Every field is a principal ideal domain.

Proof Let $I◃F$ be an ideal. If $I$ is non-trivial, then it contains a non-zero element $a$. Since $F$ is a field, $a$ is a unit, and thus $I=F$. So any field only has two trivial ideals: $\{0\}$ and $F$. $\square$

Proposition

$F[x]$ is a principal ideal domain for any field $F$.

Proof

Euclidean Algorithm

The gcd of polynomials $f,g\in F[x]$, and the linear combination that gives it can be found by the following algorithm. Compute $g=q_{1}f+r_1,f=q_2{r}_1+r_2,\cdots ,r_{n-3}=q_{n-1}{r}_{n-2}+r_{n-1},r_{n-2}=q_n{r}_{n-1}+r_n,r_{n-1}=q_{n+1}{r}_n.$Then up to a unit $r_n$ is the gcd and backward substitution gives the linear combination.

Proposition

In an integral domain every prime is irreducible.

Proof Suppose $p$ is a prime with $p=ab$. Then $p\mid ab$ implies $p\mid a$ or $p\mid b$. Without loss of generality, suppose $p\mid a$. By the equation, $a\mid p$ as well, so $a$ and $p$ are associates, thus $b$ is a unit. $\square$

Nilpotent

Let $R$ be a ring. $x\in R$ is nilpotent if there is $n\in \mathbb{N}$ such that $x^n=0$.

Def Idempotent Let $R$ be a ring. $x\in R$ is idempotent if $x^2=x$.

Proposition

If $x\in R$ is nilpotent then $1+x$ is invertible.

Recall the fundamental theorem of arithmetic:

Every integer greater than $1$ can either be prime or represented uniquely as a product of prime numbers.

Link to original

Similarly for polynomials:

Theorem

Let $F$ be a field. Every non-constant polynomial in $F[x]$ can be written uniquely (up to reordering) as $f=u{p}_1^{e_1}\dots p_r^{e_r}$for a unit $u\in F^{\times }$ and $r\in \mathbb{N}$, $p_i\in F[x]$ monic irreducible and $e_i\in \mathbb{N}$.

Both theorems are in fact one theorem:

Unique Factorization in A PID

Every principle ideal domain is a unique factorization domain.