Commutative Ring Theory/Divisibility and principal ideals

Definition (principal ideal):

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

Proposition (characterisation of divisibility by principal ideals):

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

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

Definition (similarity):

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

Proposition (similarity is an equivalence relation):

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

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

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

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

Proof: The equation ${\displaystyle \langle a\rangle =\langle b\rangle }$ implies that ${\displaystyle a=xb}$ and ${\displaystyle b=ya}$ for certain ${\displaystyle x,y\in R}$. Hence, ${\displaystyle a=xya}$. By cancellation (which is applicable because ${\displaystyle R}$ is an integral domain), ${\displaystyle xy=1}$ and hence ${\displaystyle x}$ is a unit, so that ${\displaystyle a}$ and ${\displaystyle b}$ are similar. ${\displaystyle \Box }$