We don't divide by zero because it's boring.

You can totally divide by zero, but the ring you get when you do that is the zero ring, and it only has one element. When you start with the integers and try dividing by nonzero stuff, you can say "you can't do that" or you can move out of the integers and into the rationals, into which the integers embed (or you can restrict yourself to only dividing by some nonzero things - that's called localization - which is also interesting). The difference between doing that and dividing by zero is that nothing embeds into the zero ring (except the zero ring). It's not that we can't study it, but that we don't want to.

Also, in the future, if you want to ask math questions, ask them on math.stackexchange.com (I've answered a version of this question there already, I think).