Commutative rings of characteristic 2

Is there a "canonical" form to write the commutative rings of characteristic $2$?

For instance, I hoped that every such ring is isomorphic to ${\cal P}(X)$ with symmetric difference as addition and intersection of sets as multiplication -- but quid's comment below shows that I'm mistaken in that belief. Nevertheless, ${\cal P}(X)$ has some "nice canonical twang" to it, so I still hope it can be involved in some description of rings of characteristic $2$.

• Your 'for instance' cannot be true since this would imply that there is a unique ring of characteristic $2$ for each cardinality. This is not the case already for $4$. Consider the finite field and the direct product of two fields of cardinality $2$/
– quid
Aug 6, 2015 at 13:34
• Thanks - I will edit my post to include your comment. Aug 6, 2015 at 14:05

Considering that $\mathcal{P}(X)$ with the operations you describe contains zero divisors when $|X|>1$, you will be at a loss to describe any domain of characteristic two, for example.
Commutative rings of characteristic $2$ can be quite pathological (just think of what you can do with quotients of multivariate polynomial rings over $\Bbb F_2$.)