Expanding a bit on what Chronos and ultrafilter said: If you’re talking about 0 you’re probably talking about some system with addition, since 0 means the identity for addition. If you’re talking about the possibility of division, then you’re probably talking about a system with multiplication. So, likely, you’re interested in the sort of system mathematicians call a ring, which has both addition and multiplication, behaving in nice ways (including obeying the distributive laws). In a ring you must have 0a = 0 for every element a.
Now you say that you want to be able to divide by 0. What does that mean? If you mean you want to be able to divide every element b by 0, then you want to always be able to solve 0x = b for x. But 0x is always 0, so b must always be 0. There’s one way to do this: Have 0 be your only element. It works. It’s not terribly interesting, but it works. (Extra credit: 0[sup]-1[/sup] = 0. What’s the multiplicative identity “1”?)
This isn’t the only way to make some sense out of dividing by 0, but the point is that you have to give up some familiar fact about the way numbers behave in order to do it.