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Consider the set $S = \{0,1\} $ under XOR.
It is the case that $ x \oplus x = 0 \ \ \forall x \in S$.
I was wondering if there is a similar operation in some other domain such that $x*x*x = 0$ for all objects in the domain. Can we possibly generalize this?

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  • $\begingroup$ Surely there must be some other restrictions you want to put on the operation; otherwise, the constant operation (with result $0$) qualifies. $\endgroup$
    – Brian Tung
    Jun 8, 2016 at 23:28
  • $\begingroup$ What do you mean by "constant operation" ? And what restrictions do you suggest to make the problem interesting? $\endgroup$
    – user308485
    Jun 8, 2016 at 23:39
  • $\begingroup$ I mean, if $x * y = 0$ for all $x, y$, that trivially satisfies your condition, right? But I don't think that's what you were looking for. So I assume you must want some restrictions on the operation that preclude that trivial answer. $\endgroup$
    – Brian Tung
    Jun 8, 2016 at 23:44
  • $\begingroup$ There is another function over $S$ that satisfy this. Let $0*0 = 1$ and $x*y = 0$ otherwise. $\endgroup$
    – Winther
    Jun 9, 2016 at 0:01
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    $\begingroup$ The collection of strictly lower triangular $3 \times 3$ matrices form an algebra which satisfy $x*x*x = 0$. In general, if $A$ is a strictly triangular $n\times n$ matrix, then $A^n = 0$. $\endgroup$ Jun 9, 2016 at 0:30

1 Answer 1

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If $a$ is an element in the ring of polynomials $\mathbb{Z}_3[x]$, then $a+a+a=0$. This can be generalized to characteristic $p$, where $p$ is any prime.

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  • $\begingroup$ This is the right answer. The example in the OP is an example of the general case. $\endgroup$ Jun 9, 2016 at 3:56
  • $\begingroup$ In $\mathbb{Z}_n[x]$ ($n \ge 2$), $a+a+\cdots+a$ ($n$ times) is equal to 0. If the coefficient ring $\mathbb{Z}_n$ need not be a field, then $n$ need not be a prime. $\endgroup$ Jun 9, 2016 at 5:14

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