Principal Ideal Domains

Principal Ideal Domains

Recall from the Integral Domains page that a ring $(R, +, \cdot)$ is said to be an integral domain if $R$ is a commutative ring and contains no zero divisors where we define a zero divisor in a ring to be a nonzero element $a \in R$ such that there exists a nonzero element $b \in R$ with $ab = 0$.

We now define a special type of integral domain known as a principal ideal domain.

Definition: Let $(R, +, \cdot)$ be an integral domain. Then $R$ is said to be a Principal Ideal Domain if every ideal $I \subseteq R$ is a principal ideal (that is, $I$ can be generated by a single element in $R$).

One of the simplest examples of a principal ideal domain is the integral domain $(\mathbb{Z}, +, \cdot)$. Note that if $I$ is an ideal of $\mathbb{Z}$ then $I$ must satisfy the following three properties:

  • 1) $0 \in I$.
  • 2) For all $x, y \in I$ we have that $(x + y) \in I$.
  • 3) For all $x \in I$ and for all $y \in R$ we have that $x \cdot y \in I$.

The only ideals of $\mathbb{Z}$ are $n\mathbb{Z} = \{ nx : x \in R \}$. In particular, the ideals of $\mathbb{Z}$ are:

(1)
\begin{align} \quad 0\mathbb{Z} & = \{ 0 \} \\ \quad 1\mathbb{Z} & = \mathbb{Z} \\ \quad 2\mathbb{Z} & = \{ 0, \pm 2, \pm 4, ... \} \\ \quad 3\mathbb{Z} & = \{ 0, \pm 3, \pm 6, ... \} \\ & \vdots \\ \quad n\mathbb{Z} &= \{ 0, \pm n, \pm 2n, ... \} \\ & \vdots \end{align}

Notice that for each $n \in \mathbb{Z}$, the ideal $n\mathbb{Z} = <n>$ where:

(2)
\begin{align} \quad <n> = \{ nx : x \in \mathbb{Z} \} \end{align}

Therefore every ideal in $\mathbb{Z}$ is a principal ideal. So $(\mathbb{Z}, +, \cdot)$ is a principal ideal domain.

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