Given a ring $R$, we may define an equivalence relation $a \sim b$ iff $a = ub$ for some unit $u$ (i.e. iff $a$ and $b$ are associates).
We can then consider the quotient $R / \sim$. This quotient is of course not a ring as it is not well-behaved with respect to addition, but it still seems to me like something worth studying about a ring.
My question is simply whether $R / \sim$ defined this way has a name, special properties, is studied by mathematicians, etc.
Some Observations About $R / \sim$
As mentioned above, $R / \sim$ is not a ring because it is not well-behaved with respect to addition. That is, given $[r], [s] \in R / \sim$ with $r, r' \in [r]$ and $s, s' \in [s]$, we do not have $r + s \sim r' + s'$. However, if $R$ is commutative (which we should perhaps assume to begin with) we do have $rs \sim r's'$. In other words, we can define $[r] [s] = [rs]$.
$\mathbb{Z} / \sim$ can be identified with the nonnegative integers.
$R$ is a field iff $R /\sim \, = \{[0], [1]\}$.
If $F$ is a field, $F[x] / \sim$ is the set of monic polynomials
It is similar to the definition of a projective space. Here the "scalars" belong to $R^\times$ and the "vector space" is $R$.
$R^\times$ maps to $[1]$ under the quotient.
It seems to be the case that $(\mathbb{Z} / n \mathbb{Z}) / \sim$ is precisely $\{[d] \mid d | n\}$, but the proof escapes me.