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Almost all books that I have found deal with derivation on several types of rings (or algebras) (for instance, commutative, noncommutative, domains, non-domains etc).

However, each paper about locally nilpotent derivations (that I know) suppose the ring is a domain.

Question: what happens with rings containing zero divisors and the study of locally nilpotent derivations? Does exist any phenomenon on them?

I appreciate any reference.

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Yes: Jeffrey Bergen and Piotr Grzeszczuk have co-authored a few papers on skew polynomial rings $R[x;\sigma,\delta]$ and skew power series rings $R[[x;\sigma,\delta]]$ where $\delta$ is locally nilpotent. They don't always assume $R$ is a domain: sometimes they simply require $R$ to contain a field. Here are the first few papers of theirs that I'm familiar with:

  1. On rings with locally nilpotent skew derivations, Comm. Alg., 39 (2011): 3698 -- 3708
  2. Skew derivations and the nil and prime radicals, Colloquium Mathematicum, 128 (2012): 229 -- 236
  3. Skew power series rings of derivation type, Journal of Algebra and Its Applications, 10(6) (2012): 1383 -- 1399

and there are a bunch more published after these dates that I haven't read.

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  • $\begingroup$ Are you sure that they explicitly consider LNDs on a ring with zero divisors? It seems that they pass to domains at some point. $\endgroup$
    – Binai
    Commented Apr 7, 2020 at 4:43

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