Why are Quotient Rings called Quotient Rings? Let $R$ be a ring, and $I$ be an ideal of $R$. Let $a\in R$.

Definition 1.1 : The coset of $I$ with respect to $a$ is defined to be $a+I=\{a+x:x \in I\}$
Definition 1.2 : The set of cosets of $I$ in $R$ is defined to be $R/I=\{a+I:a \in R\}$ with $+,\cdot$ defined on $R/I$ as shown.

The set $R/I$ along with operations $+, \cdot$ is called the quotient ring of $R$ by $I$ (also referred to as $R \mod I$ ).
I really don't see why we would call such a ring the quotient ring or write $R/I$. In my head this is suggestive of some kind of division of the ring $R$ with the ideal $I$, same goes for referring to it as $R \bmod I$.
Could anyone explain why we refer to this particular ring in these ways and show me how the name quotient is appropriate.
Thanks.
 A: More generally, given an equivalence relation $\sim$ on a set $X$, the set of equivalence classes is called the quotient of $X$ by $\sim$ and denoted $X/\!\sim$.
$R/I = R/\!\equiv$ where $\equiv$ is the equivalence relation given by $a \equiv b$ iff $a-b \in I$.
$\equiv$ is an equivalence relation iff $I$ is an additive subgroup of $R$.
$\equiv$ induces a ring structure on $R/I$
iff $\equiv$ is compatible with multiplication
iff $I$ is an ideal of $R$.
Reciprocally, an equivalence relation $\sim$ induces a ring structure on $R/\!\sim$ iff it is compatible with the ring structure of $R$; it is then called a congruence on $R$. In this case, $R/\!\sim=R/I$, where $I=[0]$.
A: It seems that "quotient" in this sense first appeared in the context of groups.  The quotient of a group $G$ by its normal subgroup $H$ consists of the cosets
$gH$ for $g \in G$.  If the group $G$ has $n$ elements and the subgroup $H$ has $m$ elements, then of course $m$ is a divisor of $n$ and $G/H$ has $n/m$ elements.
A: A good way to understand the motivation for the name "quotient ring" is to consider the size of the cosets in a quotient ring. To take the quotient of a ring is essentially to divide $R$ up into cosets which contain the same number of elements as the particular ideal, $I$, used to construct $R/I$.
Since the notion of quotient groups preceded that of quotient rings, the fact that $|G/N| = \frac{|G|}{|N|}$ for a finite group $G$ and a normal subgroup $N$ by Lagrange's Theorem certainly played a role.
