Principal Ideals in Integral Domain

Theorem
Let $\left({D, +, \circ}\right)$ be an integral domain.

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

Let $\left({x}\right)$ be the principal ideal of $D$ generated by $x$.

Let $x, y \in \left({D, +, \circ}\right)$.

Then:
 * $(1): \quad x \backslash y \iff \left({y}\right) \subseteq \left({x}\right)$
 * $(2): \quad x \in U_D \iff \left({x}\right) = D$
 * $(3): \quad x$ and $y$ are associates iff $\left({x}\right) = \left({y}\right)$.

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

Then by definition of divisor,

Conversely:

So $x \backslash y \iff \left({y}\right) \subseteq \left({x}\right)$.

Proof of Unit Equivalence
Conversely:

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

Then: