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 $m\in 1,2,...,2^q-1$.

It is well known that to every Boolean ring $ℛ$ can be assigned a Boolean algebra $ℬ$ whose operations are term operations of $ℛ$. Then a symmetric difference of $ℬ$ together with the meet operation recover the original ring operations of $ℛ$. The aim of this paper is to show for what a ring $ℛ$ a similar construction is possible. Of course, we do not construct a Boolean algebra but only so-called lattice-like structure which was introduced and treated by the authors in a previous paper. In particular, we reached...

De Morgan quasirings are connected to De Morgan algebras in the same way as Boolean rings are connected to Boolean algebras. The aim of the paper is to establish a common axiom system for both De Morgan quasirings and De Morgan algebras and to show how an interval of a De Morgan algebra (or De Morgan quasiring) can be viewed as a De Morgan algebra (or De Morgan quasiring, respectively).