There are many results concerning the commutativity of rings satisfying a polynomial equation. I want to know if there is any result/reference about (finite) rings that satisfy the polynomial equation $x^4=x^2$, which seems to be much more complicated than equations like $x^n=x$, etc.



Rings satisfying the polynomial equation $x^4=x^2$
Alfred Foster introduced a notion of Boolean-like ring in a 1946 paper <http://www.ams.org/tran/1946-059-01/S0002-9947-1946-0015045-5/S0002-9947-1946-0015045-5.pdf> He calls elements of a ring that satisfy $x^4=x^2$ weakly idempotent. Boolean-like ring is a commutative ring of characteristic two with identity in which $(1— a)a(1— b)b=0$ holds for all elements $a,b$ of the ring. The following properties of Boolean-like rings are well known: Each element is weakly idempotent; The nilpotent elements form an ideal;
The idempotent elements form a subring; Each element can be uniquely written as the sum of an idempotent element and a nilpotent element.


Omitting the commutativity and the existence of identity in Boolean-like rings, 
Iwao Yakabe defines generalized Boolean-like ring as a ring of characteristic two and in which $(a—a^2)(b—b^2)=0$ holds for all elements $a,b$ of the ring:
<http://catalog.lib.kyushu-u.ac.jp/handle/2324/1449033/13_2_p079.pdf>
Each element of a generalized Boolean-like ring is also weakly idempotent.


The notion of (m,n)-Boolean ring ($m>n\ge 1$) was introduced by Maurer and Szigeti as a ring in which every element satisfies $x^m=x^n$ in <https://eudml.org/doc/229780> It is claimed in the paper that the structure of (m,n)-Boolean rings heavily depends on the parity of the difference $m-n$: if this difference is odd, some reduction theorems are proved in the paper. In the case of even $m-n$ difference, no such reduction theorems are expected.


The ring of $2\times 2$ upper-triangular matrices over a Boolean ring is is an 
example of a (4,2)-Boolean ring which is not commutative.


