Talk:Nilradical of a ring

I think that the very first line is not correct. The Nilradical of the ring A is the set of nilpotents of A. It's an ideal of A but if in general it is not a nilpotent ideal (which I assume means that it exists an integer n such that the ideal to the power n is the zero ideal). It's indeed a nilpotent ideal if A is Artinian, as is written in the article. — Preceding unsigned comment added by 134.58.253.57 (talk) 14:26, 11 October 2010 (UTC)

The current article should be correct. -- Taku (talk) 11:04, 26 April 2013 (UTC)

When is the nilradical a prime ideal?

Aside from the trivial case (integral domain). -- Taku (talk) 11:01, 26 April 2013 (UTC)

The anwser is pretty obvious: it happens iff the ring has a unique minimal prime ideal. If $P\cap Q$ is prime for two prime ideals P and Q, then one of them must be contained in the other. Rschwieb (talk) 17:50, 26 April 2013 (UTC)

Yes, but I was wondering if there is any specific type of a ring; analogous to "the Jacobson radical is maximal iff the ring is local." -- Taku (talk) 18:12, 26 April 2013 (UTC)

Oh, well you just had to say so! I remember that I helped to edit Irreducible_ring a little while back, and it contained a blurb to the effect that this property is equivalent to having a unique minimal prime ideal. (Previously, such rings were the main topic of the article, but I made the terminology more descriptive with "meet" so that the article could accomodate more common uses of "irreducible ring".)

At the time I remember not knowing for sure if that claim in the article was correct, so I went searching for references. I thought I managed to unearth some resources using the terminology and justifying the claim, but right now I am failing to repeat the result! One thing I did turn up is question 7 here. I believe in the sources I found, this was the justification for calling them "irreducible rings". Rschwieb (talk) 20:25, 26 April 2013 (UTC)

I think you have answered my question. Minimal prime ideals correspond 1-1 to the irreducible components; so, as you said, the nilradical is prime iff the spectrum of the ring is irreducible. I'm not sure about the terminology, though. "irreducible ring" sounds like a ring that is irreducible as a module. Anyway, thanks. -- Taku (talk) 11:25, 29 April 2013 (UTC)

Examples

This article needs an Examples section. Vstephen B (talk) 22:43, 12 March 2023 (UTC)

Not sure that this is really needed. I have linked the lead to Radical of an ideal. There are examples there, and one gets example of nilradicals by taking, for each such example, the quotient of the ring by the ideal. D.Lazard (talk) 09:31, 13 March 2023 (UTC)