We study unitary rings of characteristic 2 satisfying identity $x^p=x$ for some natural number p. We characterize several infinite families of these rings which are Boolean, i.e., every element is idempotent. For example, it is in the case if $p=2^n-2$ or $p=2^n-5$ or $p=2^n+1$ for a suitable natural number n. Some other (more general) cases are solved for p expressed in the form $2^q+2m+1$ or $2^q+2m$ where q is a natural number and $m\in 1,2,...,2^q-1$.