Talk:Zero divisor
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I am translating math articles from the english Wikipedia into the german one, and I think the last example of this article is not correct, and that the given definition is only applicable to commutative rings.
In a noncommutative ring, a can be a left zero divisor (ab=0), right zero devisor (ba=0) or (both sided) zero divisor (ab=ca=0).
The given infinite matrix A is a left zero divisor, as A * (1&0&0&...\\ 0&0&0...\\...) = 0, but it is not a right zero divisor, because C*A=0 implies C=0. In contrast to that, B is not a left zero divisor, but a right zero divisor (using the same matrix as for A as the left factor).
Also it is only proven there that a left zero divisor cannot have a left inverse (a'a=1), but it may still have a right inverse (aa'=1). (As the example shows, the matrix A has such an inverse B.)
Can the product of two left zero divisors be not a left zero divisor? -- SirJective 10:38, 12 Aug 2003 (UTC)
- The ring Z of integers does not have any zero divisors,
What about... zero?