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.
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1$\begingroup$ Do you assume rings to be associative? unital? $\endgroup$– YCorCommented Feb 2, 2015 at 13:59
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2$\begingroup$ @YCor: usually rings are assumed to be associative and unital, but not necessarily commutative. I would assume the OP has that definition in mind.. $\endgroup$– QfwfqCommented Feb 2, 2015 at 15:20
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1$\begingroup$ I know people for which every ring is associative, unital, and commutative. I know people for which every ring is associative and unital, but not necessarily commutative. I know people for which rings are not assumed with any of these properties. In each of these communities, some people are convinced that their conventions are universal. Nevertheless I think it is useful to spare the reader from guessing which convention they use, it's really not a big effort. $\endgroup$– YCorCommented Feb 2, 2015 at 18:22
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1$\begingroup$ It isn't too hard to show that $2xy = 2yx$ for any $x,y$ in such a ring (it follows quickly from $2x^3 = 2x$). $\endgroup$– zebCommented Feb 2, 2015 at 19:09
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1$\begingroup$ Taking differences three times ($(x+1)^4-x^4 = (x+1)^2-x^2$, etc.) gives $24x=0$ for all $x$. EDIT: Actually, just taking $x=2$ gives the stronger claim $12=0$. $\endgroup$– Terry TaoCommented Feb 2, 2015 at 20:05
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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.
answered Feb 2, 2015 at 14:10
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$\begingroup$ Thank you so much! It is exactly what I was looking for. $\endgroup$ Commented Feb 2, 2015 at 23:32