In a finite ring every left-sided zero divisor is a right-sided zero divisor?

Question

A left-sided zero divisor in a ring is a non-zero element $a$ for which there is a a non-zero $b$ such that $a\cdot b = 0$. In a finite ring is it true that we can find a non-zero $c$ such that $c \cdot a = 0$ and vice-versa?

(What I'm thinking to do is constructing the element $c$ somehow, because the ring is finite, but I'm not sure how.)

One may correctly read the duped question as "Every nonzero element of a finite ring is a unit or two-sided zero divisor." The proof shows that you can't have something that is a zero divisor only on one side. — rschwieb, Jun 18 '18 at 18:03

Kenny Lau · Answer 1 · 2018-06-18T17:37:22.673

5

If there is no nonzero $c$ such that $ca=0$, then the additive group homomorphism $x \mapsto xa$ is injective (check the kernel), so it is also surjective (by finiteness), so there is $d$ such that $da = 1$.

If $ab=0$, then $dab=0$, so $b=0$.

edited Jun 18 '18 at 17:37

answered Jun 18 '18 at 17:32

Kenny Lau

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But what about left- and right-sided zero-divisors ? – Peter Jun 18 '18 at 17:34
This shows that if $a$ is not a left zero-divisor, then $a$ is not a right zero-divisor. – Joshua Mundinger Jun 18 '18 at 17:36

In a finite ring every left-sided zero divisor is a right-sided zero divisor?

1 Answers1