Mon Jan 22

The category Ab of abelian groups is an "additive category".
The inclusion Ab -> Grp has a left adjoint, "abelianization", G |-> G/G'.

Categorical def: a ring is a 1-object additive category.
Unpacked, we get the usual definition of ring-with-unit.

Def: additive functor. Consequently, we get a def: ring homomorphism. (Preserving unit!)

Some examples of rings: M_n(C), Z[i], Fun(X -> Reals), Z[x-hat, d/dx].