What is the difference between ring $\mathbb{Z}_n$ and congruence to modulo $n$. What is the difference between ring $\mathbb{Z}_n$ and congruence to modulo $n$.
Are they same thing or what are significant differences between them except we can use integers greater than $n$ for modulo operations. They look like the same.
 A: First, review the definition of what a ring is.
In particular, it's a type of algebraic structure with two different binary operations defined on it (sometimes referred to as "addition" and "multiplication").
Next, review the definition of the ring $\mathbb{Z}_n$ and you will see that the two binary operations on it are just addition mod $n$ and multiplication mod $n$.
These are interrelated concepts, but the former is a type of algebraic structure with operations, and the latter ($+$ mod $n$ and $\times$ mod $n$) are examples of operations; so they are certainly not one and the same. 
A: They are of one, but they are different kinds of mathematical objects.
Congruence modulo $n$ is an equivalence relation on $\Bbb Z$, whereas $\Bbb Z_n$ is a ring, namely the quotient ring by the congruence modulo $n$ relation (meaning that two elements of $\Bbb Z$ are regarded identical in $\Bbb Z_n$ iff they are congruent mod $n$).
A: To elaborate on my comment, something positive can be said about the relation between the congrence modulo $n$ and the ring  $\Bbb Z/n\Bbb Z$: the latter is the quotient of the the ring $\Bbb n$ by the relation of congrence modulo $n$.
This means the elements of $\Bbb Z/n\Bbb Z$ are classes of the relation of congrence modulo $n$ (which is an equivalence relation; any equivalence relation gives a partition of the set it is defined on into equivalence classes). Moreover the operations of addition and multiplication in $\Bbb Z/n\Bbb Z$ are derived from those operations in $\Bbb Z$: to perform one of these operations, one chooses elements (representatives) from the classes to be added/multiplied, one performs the corresponding operation upon them in $\Bbb Z$, and finally one takes the class (modulo $n$) of the resulting number.
It is a non-trivial fact, which can be proved, that every set of choices of representatives will give the same end result (a class modulo $n$); this is why the quotient ring of $\Bbb Z$ by the relation of equivalence modulo $n$ is defined in the first place. (And it explains why there is no such thing as a quotient ring $\Bbb R/n\Bbb R$; in this case the required property for defining multiplication fails.)
