3

Take the lattice of ideals in a non-commutative ring with 1. It is well known that the lowerbounds of all the maximal ideals are the superfluous ideals. Is there a similar characterization for the lowerbounds of all the prime ideals? They must be close to the sum of nilpotent ideals but I can't seem to prove it.

Chrystomath
  • 31
  • Two-sided ideals? How are you defining "superfluous ideal"? An ideal $S$ for which $I+S=R$ implies $I=R$? – rschwieb May 23 '13 at 18:06

1 Answers1

2

The situation for the maximal ideals is clear cut since the greatest lower bound (their intersection, the Brown-McCoy radical) happens to be the biggest superfluous ideal of the ring. Any ideal below it is also superfluous, and it contains all superfluous ideals.

Following this, we consider the intersection of all prime ideals (The "lower nilradical", "prime radical" or "Baer radical").

It's known (you can check Lambek's Lectures on rings and modules) that the prime radical of a ring consists of all its strongly nilpotent elements. (The definition of $a$ being "strongly nilpotent" is that every sequence beginning with $a_1=a$ such that $a_{i+1}\in a_nRa_n$ has a term which is zero, so that it terminates in zeros.)

If we define a "strongly nil ideal" to be one in which all elements are strongly nilpotent, then the prime radical is the largest strongly nil ideal, and it is obvious that all ideals contained within it are strongly nil ideals, and it's clear that all strongly nil ideals are contained in the prime radical.

The sum of all nilpotent ideals is obviously contained inside the prime radical, but it may be strictly smaller.

rschwieb
  • 153,510
  • I read this great Classic book.+1 for reminding me that. :-) – Mikasa May 27 '13 at 11:29
  • 1
    @rschweib Can you provide an example of a ring for which the sum of nilpotent ideals is not semiprime? – Dustan Levenstein Nov 15 '16 at 02:55
  • @DustanLevenstein None are coming to mind for me. Interesting challenge – rschwieb Nov 15 '16 at 15:54
  • In case it wasn't clear, this is a direct reference to your last sentence - "The sum of all nilpotent ideals is obviously contained inside the prime radical, but it may be strictly smaller." – Dustan Levenstein Nov 15 '16 at 16:47
  • @DustanLevenstein at first glance I thought your question was strictly harder, but yes, of course it is equivalent to exhibiting an example for that. I'll let you know if I come across anything. I haven't thought about it in years – rschwieb Nov 15 '16 at 16:59
  • Lol, then I won't feel bad for not having been able to come up with an example myself. :) – Dustan Levenstein Nov 15 '16 at 18:02
  • @DustanLevenstein While looking for some properties that might help prove or disprove that this is possible, I ran across this which looks really informative. Thought you might like it too. – rschwieb Nov 15 '16 at 18:09