Ring of Residue Classes Modulo $n$

While the above description gives a complete definition of this ring, it can also be defined equivalently as the quotient ring of the $m\mathbb{Z}$ ideal in the integers, that is

In this case, we identify each element by a coset of $n\mathbb{Z}$ of the form $a+n\mathbb{Z}$, defining addition and multiplication as is done in quotient rings in general

In this representation, it is easy to also see that representatives fall in the same coset if they differ by a multiple of the ideal generator

This representation shows why the original definition does indeed define a ring.