The rings which are Boolean

We study unitary rings of characteristic 2 satisfying identity ${\displaystyle x^p=x}$ 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 ${\displaystyle p=2^n-2}$ or ${\displaystyle p=2^n-5}$ or ${\displaystyle p=2^n+1}$ for a suitable natural number n. Some other (more general) cases are solved for p expressed in the form ${\displaystyle 2^q+2m+1}$ or ${\displaystyle 2^q+2m}$ where q is a natural number and ${\displaystyle m\in \{1,2,...,2^q-1\}}$.