0/0 exists in the 0 ring

{0} is a ring under multiplication. 0 is the multiplicative identity and additive identity.

we have 0 x 0 = 0
so 0/0 = 0 and everythign works fine!

>>1
Here's your problem. Sure your ring can have multiplication as one of it's operations, but that doesn't mean division is defined. And if division -is- defined, then it's not the same division that we'd normally use to divide real number quantities. Simply put, division is only defined on Rx(R-{0}), assuming that (a,b)=a/b. So, in order to have division on your ring, you're going to have to define an operation other than the one we're all familiar with.

Have fun with your new operation :P