Let $R$ be a ring with $1$. We say that $a,b\in R$ are associate if $a|b$ and $b|a$, i.e. there exist $r,s\in R$ such that $a=rb$ and $b=sa$. This is an equivalence relation on $R$, and I would like to know if the quotient set $R/\sim$ (where $a\sim b$ when $a$ and $b$ are associate) has interesting properties such that a given structure or information about $R$.
In particular, I'm interested in the case where $R$ is an integral domain, where $a\sim b$ if and only if $r$ and $s$ above are units. If further the units of $R$ commute with every element, then $R/\sim$ inherits a monoid structure from $R$, since $ua\cdot vb=uvab$, and therefore the product in $R$ induces a well defined product in the quotient. However, the sum in $R$ isn't preserved in general in $R/\sim$.
Some examples I've played with without getting much more information are $\mathbb{Z}$ (where the monoid is just $\mathbb{N}$), fields (monoid $\mathbb{Z}/(2)$), several instances of $\mathbb{Z}/(n)$ (I know these are not in general integral domains, but I wanted to know how they look like) and some polynomial rings.
To sum up, my question is
Has this quotient set been studied for some family of rings and has any interesing properties?
