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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 $2$.

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    $\begingroup$ 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 $2$/ $\endgroup$ – quid Aug 6 '15 at 13:34
  • $\begingroup$ Thanks - I will edit my post to include your comment. $\endgroup$ – Dominic van der Zypen Aug 6 '15 at 14:05
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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.

What's more, the multiplication you're describing on the powerset makes the ring (and all of its subrings) a Boolean ring, which is only a small subclass of rings of characteristic 2.

Rings of sets do come into play in the study of Boolean rings. Perhaps you're already aware of that, but if you aren't, look into Stone's theorem.

Commutative rings of characteristic $2$ can be quite pathological (just think of what you can do with quotients of multivariate polynomial rings over $\Bbb F_2$.)

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