Principal Ideals in Integral Domain

Theorem
Let $\struct {D, +, \circ}$ be an integral domain.

Let $U_D$ be the group of units of $D$.

Let $\ideal x$ be the principal ideal of $D$ generated by $x$.

Let $x, y \in \struct {D, +, \circ}$.

Then:
 * $(1): \quad x \divides y \iff \ideal y \subseteq \ideal x$
 * $(2): \quad x \in U_D \iff \ideal x = D$
 * $(3): \quad x$ and $y$ are associates $\ideal x = \ideal y$.

Proof of Divisor Equivalence
Suppose that $x \divides y$.

Then by definition of divisor,

Conversely:

So:
 * $x \divides y \iff \ideal y \subseteq \ideal x$

Proof of Unit Equivalence
Conversely:

Proof of Associatehood
Let $x \cong y$ denote that $x$ and $y$ are associates.

Then: