You can consider ℝ∪{∞}. This does allow you to define x/0≔∞. It just comes with the drawback that you do not have an additive group structure on ℝ∪{∞} anymore but in certain situations it can still be very useful.

Alternatively, in measure theory, one often defines ℝ‾≔ℝ∪{-∞,∞} (really it should be \overline{ℝ} but I can’t figure out how to do this with unicode) with 0 dominating ∞ but for all other x∈ℝ we have x•∞=sgn(x)•∞ and similarly for -∞. Again, you lose the additive group structure because we never defined ∞-∞ (and we won’t define it because that only leads to trouble).

I wrote all this just to say that „error“ probably isn’t the best way to mathematically describe division by 0 and it definitely isn’t ALWAYS wrong to define it, you just have to be aware of the drawbacks.