Talk:Zero divisor

message: The article claims that the set of zero divisors of a commutative ring is a prime ideal. This is wrong.
Neither the product nor the sum of zero divisors need be zero divisors. In fact, if a,b are zero divisors
different from zero, then there are r,s different from zero such that ar=0 and bs=0. Maybe the author
meant that (ab)(rs)=0 and hence ab should be a zero divisor. However, it may happen that rs=0. Therefore
the conclusion, that ab is a zero divisor, does not follow from this. 
Anyway, it is true that the set Z of zero divisors is a union of prime ideals, and it contains
every minimal prime ideal. This is due to the fact that the complement of Z is a saturated
multiplicatively closed set (and contains all units). ASlateff 128.131.37.74 16:24, 4 January 2007 (UTC)
replies: text: If we use the definition of zero divisor that allows 0 to be a zero divisor, then any multiple of a zero divisor in a commutative ring is also a zero divisor. The obvious non-commutative counter-example are zero divisors x and y such that xy = 1. Albmont (talk) 18:22, 22 January 2008 (UTC)
replies: text:  But if xy = 1, then x and y are units, and hence not zero divisors.  (The proof in the article that zero divisors cannot be units did not rely on commutativity, so it should hold for non-commutative rings as well.)  Your counter-example cannot exist.  —Preceding unsigned comment added by 209.147.104.77 (talk) 14:01, 20 March 2011 (UTC)

text: The author says "Set of zero divisors is a union of prime ideals." I guess there should be "minimal prime ideals" instead of "prime ideals". Minimal prime (commutative algebra) --94.54.14.47 (talk) 14:22, 7 March 2010 (UTC)
replies: text:  That guess is incorrect: for example, in k[x,y]/(x^2,xy), the unique minimal prime is (x), but y is a zero divisor not in this minimal prime. Rschwieb (talk) 14:58, 14 November 2013 (UTC)
thread_title:  In general, zero divisors do not form a prime ideal! 

message: the first example:
(1,0)\times(0,1)=(0,0)
Seems to overlook a small detail that there is no cross product in R2, in fact the "cross product" as we know it is only defined in  R3, being a particular case of the wedge product. I took the liberty of changing the cross product to a dot product, either way, in the current example the wedge product would yield one (scalar).  —Preceding unsigned comment added by Carvasf (talk • contribs) 23:18, 7 June 2008 (UTC)
replies: text: The cited example doesn't involve cross products or dot products.  Rather, it involves componentwise multiplication, which is the standard product in the ring Z×Z.  I've edited the example a bit in an attempt to clarify this. Michael Slone (talk) 01:35, 8 June 2008 (UTC)
thread_title:  Cross product in '''R'''<sup>2</sup> 

message: In its present version, the article reads: "An element that is both a left and a right zero divisor is simply called a zero divisor". This is indeed coherent with _some_ sources (since we are Saturday, I have no library at hand but Google Books is my friends), e.g.  (A course in algebra by Yun Fan, Q. Y. Xiong, Y. L. Zheng) but certainly not all e.g. Concepts in abstract algebra by Charles Lanski  which defines a zero divisor as an element which is either a right zero divisor or a left zero divisor, and uses two-sided zero divisor for elements which are both.
The present version seems odd to me, and it seems to have propagated to most if not all Wikipedias in major languages. I have enough to do to correct the numerous mistakes in Wikipedia in French, so I just post the problem here on the discussion page, ready to have a look at the replies - but I don't believe things are rightly done in the current version. French Tourist (talk) 13:24, 5 February 2011 (UTC)
replies: text: I have added the Lanski reference and changed the article according as per this reference. — Quondum☏ 16:01, 10 August 2012 (UTC)
thread_title:  Bug on the definition of "zero divisor" from the definition of lateral zero divisors 

message: I'm curious to know why they're called "zero divisors"; since the term seems to define the case where two non-zero numbers multiply to produce zero, shouldn't they be called "zero factors" instead? The Grand Rascal (talk) 06:08, 28 September 2020 (UTC)
thread_title:  Why Is It Called "Zero Divisor"?