Elements in same orbit under multiplication by group of units are associate

Statement
Suppose $$R$$ is a commutative unital ring and $$a,b \in R$$ are elements such that there exists a unit $$u \in R$$ such that $$b = au$$. Then, $$a$$ and $$b$$ are fact about::associate elements in $$R$$.

Unit
An element $$u$$ of a commutative unital ring $$R$$ is termed a unit if there exists $$v \in R$$ such that $$uv = 1$$.

Associate elements
Two elements $$a,b$$ of a ring $$R$$ are termed associate elements if they divide each other: $$a | b$$ and $$b | a$$.

Related facts

 * Associate implies same orbit under multiplication by group of units in integral domain
 * Associate not implies same orbit under multiplication by group of units

Proof
Given: A commutative unital ring $$R$$, elements $$a,b \in R$$ such that $$b = au$$ for a unit $$u$$ of $$R$$.

To prove: $$a$$ and $$b$$ are associate elements in $$R$$.

Proof: By the definition of unit, there exists $$v \in R$$ such that $$uv = 1$$. We then have: $$bv = (au)v = a(uv) = a1 = a$$. Thus, $$b | a$$. Also, $$au = b$$, so $$a | b$$. Thus, $$a,b$$ are associate elements.