Quotient ring $\mathbb{Z}/4\mathbb{Z}$ I'm trying to understand what the quotient ring is. I know that $\mathbb{Z}/4\mathbb{Z} = \mathbb{Z}_4$, but I can't get the same result by myself. Having used the definition of the quotient ring that $R/I = \left \{x + I | x \in R \right \}$ I've got: 
$$ 4\mathbb{Z} = \left \{4x | x \in \mathbb{Z} \right \}\\
\mathbb{Z}/4\mathbb{Z} = \left \{y + 4x | x,y \in \mathbb{Z}\right \}  $$
It looks like my result equals $\mathbb{Z}$ and it isn't $\left \{0, 1, 2, 3 \right \}$. I see that I miss something very important in the concept of the factor ring. I would be very grateful if someone would explain me the quotient ring by using this example.
 A: When looking at the quotient you are identifying elements with other elements, so you are not looking at integers $y+4x$ but at classes of integers $\{y+4x:x\in \mathbb{Z}\}$. In your special case there are four such classes, one for each $y\in \{0,1,2,3\}$
(Edit: so in your definition, on the right hand side, it's not correct to write "$x,y\in \mathbb{Z}$", but it should read "x\in \mathbb{Z}" and in an 'outer' curly bracket pair '$y\in\mathbb{Z}$'. Compare with your general definition involving $R$ and $I$) 
A: Applying your definition of quotient ring, $\mathbb{Z}/4\mathbb{Z}=\{y+4\mathbb{Z}:y\in\mathbb{Z}\}=\{\{y+4x:x\in\mathbb{Z}\}:y\in\mathbb{Z}\}$.  (I like to imagine these as a collection of interlocking combs with equally spaced teeth along the number line.)
Notice this is a set of sets of integers, rather than a set of integers.  This particular kind of set of sets is also called "a collection of equivalence classes."
This does not equal $\{0,1,2,3\}$, though.  Instead, there is a bijection between $\{0,1,2,3\}$ and $\mathbb{Z}/4\mathbb{Z}$ defined by $y\mapsto y+4\mathbb{Z}$.
A: The issue is you're confusing what elements $\mathbb{Z}/4\mathbb{Z}$ has with what is written out. Yes in $\mathbb{Z}/4\mathbb{Z}$ we have all elements of $\mathbb{Z}$ but that is not what is interesting anymore, what has happened is that we've created a new way to say what elements equals to what else. As we area dealing with a quotient ring we have that $a=b$ if and only if $a-b\in 4\mathbb{Z}$
This is the important part, so we have that $3=7$ because $7-3=4\in4\mathbb{Z}$, as this goes for anything we can just use the lowest four numbers, namely $0$,$1$,$2$ and $3$ as those are the only ones not in $4\mathbb{Z}$, except for the $0$ of course. This is because any number That is equal to 3 can be written in the form of $3+4k$ with $k\in\mathbb{Z}$ and $3+4k-3=4k\in4\mathbb{Z}$ is fairly evident.
Addition and multiplication can be shown to be valid the same way using this equivalence relation it induces.
