TITLE: Characterize the category of rings
QUESTION [19 upvotes]: (Sub)categories of many well-studied mathematical objects have been characterized purely in terms of their morphisms. Some (famous) examples:

Sets and functions, due to Lawvere.
Modules over some ring and module homomorphisms, due to Mitchell and Freyd.
Sheaves over some site and natural transformations, due to Giraud.
Topological spaces and continuous functions, due to Schlomiuk.
Boolean algebras and homomorphisms, by combining the above.

In a similar vein:

Is there a characterization of the category of rings and homomorphisms?

REPLY [11 votes]: Yves Diers has the notion of a Zariski category in [Categories of commutative algebras], which apparently suffices to carry out a lot of commutative algebra in an axiomatic fashion. I reproduce the definition:

A Zariski category is a category $\mathcal{A}$ satisfying the following conditions:

$\mathcal{A}$ is cocomplete.
$\mathcal{A}$ has a strong generating set whose objects are finitely presentable and flatly codisjunctable.
Regular epimorphisms are universal i.e. stable under pullbacks.
The terminal object of $\mathcal{A}$ is finitely presentable and has no proper subobject.
Binary products of objects are co-universal i.e. stable under pushouts.
For any finite sequence of codisjunctable congruences $r_1, \ldots, r_n$ on any object with respect codisjunctors $d_1, \ldots, d_n$, we have
  $$r_1 \vee^c \cdots \vee^c r_n = \mathrm{id}_{A \times A} \implies d_1 \vee \cdots \vee d_n = \mathrm{id}_A$$
  where $\vee^c$ denotes the join in the lattice of congruences on $A$, while $\vee$ denotes the co-union of quotient objects of $A$.


For more details, see the cited book, or the introduction of this article.