Commutative Ring Theory/Divisibility and principal ideals

Definition (principal ideal):

Let $R$ be a commutative ring. A principal ideal is a left principal ideal of $R$. Equivalently, it is a right principal ideal or a two-sided principal ideal of $R$.

Proposition (characterisation of divisibility by principal ideals):

Let $R$ be a commutative ring, and let $a,b\in R$. Then $a|b\Leftrightarrow \langle a\rangle \geq \langle b\rangle$.

Proof: Both assertions are equivalent to the existence of a $c\in R$ such that $b=ac$. $\Box$

Definition (similarity):

Let $R$ be a commutative ring. Two elements $a,b\in R$ are called similar if and only if there exists a unit $u\in R$ such that $a=ub$.

Proposition (similarity is an equivalence relation):

Given a ring $R$, the relation of similarity defines an equivalence relation on the elements of $R$.

Proof: For reflexivity, use the identity, and for symmetry, use the inverse. Suppose that $a=ub$ and $b=vc$, where $u,v\in R^{\times }$. Then $a=uvc$, where of course $uv\in R^{\times }$. $\Box$

Proposition (in an integral domain, the generating element of a principal ideal is unique up to similarity):

Let $R$ be an integral domain, and let $\langle a\rangle \leq R$ be a principal ideal of $R$. Then if $\langle a\rangle =\langle b\rangle$ for some element $b\in R$, we have $a=ub$ for some $u\in R^{\times }$.

Proof: The equation $\langle a\rangle =\langle b\rangle$ implies that $a=xb$ and $b=ya$ for certain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle x, y \in R} . Hence, $a=xya$. By cancellation (which is applicable because $R$ is an integral domain), $xy=1$ and hence $x$ is a unit, so that $a$ and $b$ are similar. $\Box$