About the ring $\mathbb{Z[\sqrt{4}]}$ I recently learned about quadratic ring extensions and among the various examples we were given $\mathbb{Z}[\sqrt{4}]$.
I am a bit puzzled by this object. I understand that it is not an Integral Domain since it has zero divisors and so a lot of nice properties are lost, but I can't seem to "fit" it anywhere among the other $\mathbb{Z}[\sqrt{d}]$ (where $d$ is not a square in $\mathbb{Z})$.
Is there some context in which the ring $\mathbb{Z}[\sqrt{4}]$ arises naturally? 

As suggested in comments, this notation is fatally ambiguous. That is, if it means "adjoin a square root of 4 to $\mathbb Z$", then we "adjoin" $2$ or $-2$, and, either way, have ... depending upon interpretation... done nothing, since there was already a $2$ and/or a $-2$ in $\mathbb Z$. 
If, as comments suggest, it was meant to consider a sense in the general pattern of $\mathbb Z[x]/(x^2-4)$, then EDIT: as @rschwieb pointed out, the ideals $(x-2)$ and $(x+2)$ in $\mathbb Z[x]$ are not coprime... but let's pretend it's $\mathbb Q[x]$... (via Sun-Ze's theorem, a.k.a. "Chinese Remainder Theorem") we get $\mathbb Q[x]/(x-2)\oplus \mathbb Q[x]/(x+2)$, which is isomorphic to $\mathbb Q\oplus \mathbb Q$. With zero divisors, yes, etc. With $\mathbb Z[x]$, it's messier...
