Actually not. You are forgetting some key things. Such as limits. If we take the limit of 1/x as x approaches 0 we get infinity and here is how the magic happens. You can't do arithmetic on infinite values. It just outright fails. Therefore since your theorem is entirely arithmetic it will fail. Whereas limits are not arithmetic so my theorem standsLet me demonstrate what happens if zero had a multiplicative inverse in a ring with 1.
Suppose there was an a with 0*a = 1.
Then 0*a = 0*a + 0 = 0*a + 0*a - (0*a) = (0+0)*a - (0*a) = 0*a - (0*a) = 0.
So we have 1 = 0.
For each b in the ring we have b = 1*b = 0*b = 0. In other words, all elements of the ring are zero.
So you can only divide by zero if nothing else than zero exists.