Math and science::Algebra::Aluffi
The uniqueness of the zero ring
A ring is the zero ring iff
That is to say, the zero ring is the only ring where both operations share the same element for their identity.
The proof is on the reverse.
Proof. The forward case is trivial, as
there is only one element. So consider the reverse case. Let
By
But by the distributive law, this means that:
which can only be true for the additive identity. So