Definition of a 2-primal ring

Question

In defining a 2-primal ring, one usually means a ring such that the set of nilpotent elements equals the prime radical (the intersection of prime ideals). So, the set of nilpotent elements is readily an ideal in 2-primal rings. Is it true that, from the begining, one define a 2-primal ring as a ring in which the nilpotent elements form an ideal? Namely, are these two statements equivalent to each other?

Thanks for any contribution!

Answer 1

No, although I don't have a counterexample at hand.

It is suggested in Reversible and symmetric rings by Greg Marks, and On nil-semicommutative rings by Mohammadi, Moussavi & Zahiri that the class of rings whose nilpotent elements form an ideal is strictly larger than the class of $2$-primal rings.

I'm quite sure if this definition were equivalent, or if the question was open, they would have mentioned it. This leads me to believe there is an accessible counterexample.

I will continue searching for a counterexample, and I am very interested in seeing one if someone points to one first.

Update: I can't say that the description is yet included on the DaRT page, but it at least provides a few citations to the example:

• G. Marks. On 2-primal Ore extensions. (2001) @ Example 2.2
• G. Marks. Skew polynomial rings over 2-primal rings. (1999) @ Example 2.1
• J. Ram. On the semisimplicity of skew polynomial rings. (1984) @ Example 3.2

Answer 2

An example may be found in Example 2.2 of this paper .