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) $$x \backslash y \iff \left({y}\right) \subseteq \left({x}\right)$$;
 * 2) $$x \in U_D \iff \left({x}\right) = D$$;
 * 3) $$x$$ 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:

$$ $$ $$