I'm not a professional mathematician (nor anything remotely close to that), but for sometimes I've wondered about this.
Would it be possible to magicate some strange mathematical concept (lets call it $ S $), if that's even the correct word, such that:
$$ S \times 0 = 1 $$
Or in other words:
$$ S = \frac{1}{0} $$
I'm imagining something which at first looks absurd, such as the imaginary unit $ i = \sqrt{-1} $, which nonetheless works very well in mathematics and in the real word.
My sparse math skills can't take much further than some basic and sometimes bizarre properties such as the fact that $ S $ seems to be completely indiferente to addition/subtraction, division and exponentiation for the natural numbers:
$$ n \in \mathbb{N} $$
$$ S + n = \frac{1}{0} + n = \frac{1 + n \times 0}{0 \times 1} = \frac{1}{0} = S \Rightarrow S + n = S $$
$$ S - n = \frac{1}{0} - n = \frac{1 - n \times 0}{0 \times 1} = \frac{1}{0} = S \Rightarrow S - n = S $$
$$ \frac{S}{n} = \frac{1}{0} \times \frac{1}{n} = \frac{1 \times 1}{0 \times n} = \frac{1}{0} \Rightarrow \frac{S}{n} = S $$
$$ S^n = \frac{1^n}{0^n} = \frac{1}{0} = S \Rightarrow S^n = S $$
For multiplication it's a bit more complicated, but the result is the same and it doesn't violate any of the above (I think). Illustrating for $ n = 2 $:
$$ 2 \times S = S + S = \frac{1}{0} + \frac{1}{0} = \frac{1 \times 0 + 1 \times 0}{0^2} = \frac{1/S}{1/S} = \frac{S}{S} \Rightarrow S = \frac{S}{S} - S = S \Bigl(\frac{1}{S} - 1\Big) = S (0 - 1) = S \times n = S \Rightarrow S \times n = S$$
Would this idea break math? I feel it's the most likely case, but I'm not skilled enough to pursue this much further than the above mentioned basics. Anyway, for me it's an interesting thing to think about, and I'd love the support of the community to come up with ways in which this idea fails or violates it's own rule-set in some manner.
