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Does $ax = xa = 0$ for all $x$ in a ring imply that $ a = 0$?

I know the answer is obvious if the ring has unity, but does the claim holds for any ring?

Mengfan Ma
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    Not exactly a duplicate, but my answer there also applies : https://math.stackexchange.com/questions/1626648/is-there-any-trivial-ring-which-isnt-null/1626661#1626661 (the counterexample in the other answers are the same as the one given by RGS, by the way) – Arnaud D. Oct 18 '17 at 09:02

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$\forall x\ ax = 0\ \implies a^2 = 0$

I used this to look for a ring where $a$ is its own zero divisor, and found one:

Consider the subring of $\mathbb{Z}_4$ that consists of $\{0, 2\}$. It is true that $2x = 0$ for all $x$, but $2 \neq 0$.

RGS
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