The title says pretty much what I want. Of course, the abelian categories should contain at least one nonzero object.
In particular, is there an abelian category containing only one nonzero object? On the one hand, this is equivalent to construct a ring which is the endomorphism of the nonzero object. On the other hand, this is equivalent to construct a special module by Freyd–Mitchell theorem.
This seems silly for that's not what abelian category is invented for, but I really want to know the answer.