Talk:Reduced ring

Thank you Darij for fixing my mistake. However, the criterion

• every element $a\in A$ satisfying $a^{2}=0$ is zero itself

is also wrong: the ring Z/6Z satisfies this criterion but is not reduced. I have fixed this. Ninte (talk) 11:09, 20 October 2010 (UTC)

I did not notice this strange comment before: the condition is equivalent. The ring Z/6Z is clearly reduced. Even if one does not believe the squaring criterion, it is still obvious that 6 never divides any power of 1, 2, 3, 4, or 5 evenly. In fact, my main purpose today is to add the Z/6Z example because it is an example of a finite reduced ring, unlike the other two. Rschwieb (talk) 20:16, 29 December 2010 (UTC)

Oh yes, I agree now. Ninte (talk) 16:00, 28 February 2011 (UTC)

Proof

Suppose that $a^{n}=0$. If n is even, divide it by 2. If n is odd, add 1 to it and divide by 2. Repeat this process until n=1, and then a=0. GeoffreyT2000 (talk) 01:56, 2 May 2015 (UTC)

Origin of term

Of course, "having no nonzero nilpotent elements" is a natural condition, but I was curious as to the origin of the term "reduced" in the ring theory context. As far as I can tell, the term seems to have been first used in this way in French ("anneaux réduits") by Guy Renault in 1967:

G. Renault, Anneaux réduits non commutatifs.(French), J. Math. Pures Appl. (9) 46 (1967), 203-214.

I am fairly certain this paper is the source because (a) Renault does not cite anyone for the term, (b) in a paper written just one year earlier, Renault does not bother giving the property a name, and (c) I cannot find an earlier paper by any author that uses "reduced" in this sense. As far as I can see, the first use of "reduced" in English is a 1973 paper by Stuart Steinberg, citing Renault.

It's very tempting to insert a sentence or two somewhere in the article discussing this, but I certainly could have missed a source. So unless someone talks me into putting it in the article, I'm just mentioning it here for now. Michael Kinyon (talk) 22:32, 13 December 2025 (UTC)