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

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