Talk:Zero-product property

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Overhauled the page and removed the if and only if statement from the original: one direction always holds. Xantharius 18:31, 3 May 2006 (UTC)[reply]

Another thought: should this article not be entitled zero-product rule instead? Xantharius 18:39, 3 May 2006 (UTC)[reply]

The false statement that follows was hidden by an unknown editor. I've moved it here since it could be more useful as a warning not to repeat the error. Zaslav (talk) 06:02, 5 March 2010 (UTC)[reply]

Since the zero-divisor set Z(R) of a ring R forms a prime ideal of R (proved here), it follows that the quotient of R by Z(R), or R/Z(R) is an integral domain (also proved here in the general case). That is to say, by forming equivalence classes based on all elements on which the zero-product property does not hold, one obtains a ring structure in which the zero-product property holds for all elements in the quotient.

COUNTER-EXAMPLE: in Z6, the zero divisors are {0,2,3,4}

The box at the top of the page says: ab=0 implies a=0, b=0. This is misleading. The word "or" must be put in, since the comma could easily suggest "and." Zaslav (talk) 06:17, 5 March 2010 (UTC)[reply]

Why not merge into Zero divisors?

I hope someone will explain why this article exists. Most of it seems to belong in the article on zero divisors. The basic concept from the advanced-math viewpoint, as far as I know, is zero divisors, not "zero-product property", which is just a name for the property of not being a zero divisor. Moreover, the name "zero-product property" is not standard; in fact, as far as I know, there is no generally accepted name for this property. Usually this property of a ring is called "having no zero divisors". Does someone has something to say about this? Maybe I'm just very old fashioned. (Quotation.) Zaslav (talk) 06:08, 5 March 2010 (UTC)[reply]

At least abstractly, the zero product property described on this page could be a more general concept than zero divisors, which are explicitly tied to rings. Perhaps other algebraic structures possess basically the same property. All the examples from the page are in rings, so unless this is actually the case, I agree that having two pages is unhelpful. 67.158.43.41 (talk) 09:06, 24 March 2011 (UTC)[reply]

This page has the nice property that people like me can understand most of it despite being ignorant about abstract algebra, including being ignorant about rings. (I'm not proud of the ignorance, you understand; I'm just grateful that education about the zero product property was accessible.) The section on "Application to finding roots of polynomials" is particularly welcome in answering the always-present question "Why should I care?" In this case, I should care because the zero product property justifies the root-finding process discussed in eighth grade. So without deterring anyone from improving and merging articles, I hope that the welcome, segmented education can be maintained. DavidHolmes0 (talk) 02:08, 31 December 2013 (UTC) (p.s. Thank you, previous editors.)[reply]