In a finite ring, a unit isn’t a zero factor, vice versa.

Question

We know that in a finite ring, every non-zero element is either a unit or a zero factor. Does there exists an element is both a unit and a zero factor?

For instance, suppose that: in a finite ring, we have ab = 1 and we want to show a is not a zero factor and ba = 1.

Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. — Community, Oct 15 '24 at 09:33
It has been proved that every nonzero element in a finite ring is either a unit or a zero divisor. See click me! — ginger cat, Oct 15 '24 at 09:37

score -1 · Answer 1 · answered Oct 15 '24 at 09:42

-1

No. Suppose that $x$ is both a unit and a zero divisor in a commutative ring $R$. I.e, here exist both $w \neq 0$ and $u \neq 0$ such that $wx = 0$ and $xu=1$, then
$$w = w(xu) = (wx)u=0. $$

answered Oct 15 '24 at 09:42

ursua

7

If ab=0, ba=0? i.e. xw=0 and xu=1 – ginger cat Oct 15 '24 at 09:54
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Please don't answer problem statement questions and duplicates. https://math.meta.stackexchange.com/questions/33508/enforcement-of-quality-standards – Martin Brandenburg Oct 15 '24 at 10:14
I think my question is not the duplicate as the one listed above. My question is , in any ring, can a unit itself be a zero factor? I mean that it is not only a unit but a zero factor. – ginger cat Oct 15 '24 at 10:54
Your question has been answered there. Please read carefully. – Martin Brandenburg Oct 15 '24 at 10:57
That question used commutativity, but here we are talking about a random finite ring, so we can't have things like an inverse of both sides and we can't dirive ba = 1 from ab = 1 – ginger cat Oct 15 '24 at 11:07
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A unit has, by definition, a two-sided inverse (Wikipedia. So there is no issue. Also notice that, in any case, in a finite ring $ab=1$ always implies $ba=1$. See for example here. – Martin Brandenburg Oct 15 '24 at 11:15

In a finite ring, a unit isn’t a zero factor, vice versa.

1 Answers1