No Access

On nilpotent elements of ore extensions

Masoud Azimi

Department of Mathematics, Tarbiat Modares University, P. O. Box. 14115–170, Tehran, Iran

E-mail Address: Azimidr45@gmail.com

Search for more papers by this author

and

Ahmad Moussavi

Department of Mathematics, Tarbiat Modares University, P. O. Box. 14115–170, Tehran, Iran

E-mail Address: moussavi.a@gmail.com

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S1793557117500437Cited by:0 (Source: Crossref)

Abstract

Let $R$ $R$ be an associative ring with unity, $α$ $α$ be an endomorphism of $R$ $R$ and $δ$ $δ$ an $α$ $α$ -derivation of $R$ $R$ . We introduce the notion of $α$ $α$ -nilpotent p.p.-rings, and prove that the $α$ $α$ -nilpotent p.p.-condition extends to various ring extensions. Among other results, we show that, when $R$ $R$ is a nil- $α$ $α$ -compatible and $2$ $2$ -primal ring with $Nil (R)$ $Nil (R)$ nilpotent, then $Nil (R [x; α, δ]) = Nil (R) [x; α, δ]$ $Nil (R [x; α, δ]) = Nil (R) [x; α, δ]$ ; and when $R$ $R$ is a nil Armendriz ring of skew power series type with $Nil (R)$ $Nil (R)$ nilpotent, then $Nil (R [[x; α]]) = Nil (R) [[x; α]],$ $Nil (R [[x; α]]) = Nil (R) [[x; α]],$ where $Nil (R)$ $Nil (R)$ is the set of nilpotent elements of $R$ $R$ . These results extend existing results to a more general setting.

Communicated by L. Bokut

Keywords:

AMSC: 16S36, 16W60

References

1. A. Amitsur, Algebras over infinite fileds, Proc. Amer. Math. Soc. 7 (1956) 35–48. Crossref, Google Scholar
2. D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 7 (1998) 2265–2272. Crossref, Web of Science, Google Scholar
3. R. Antoine, Nilpotent elements and Armendariz rings, J. Algebra 319 (2008) 3128–3140. Crossref, Web of Science, Google Scholar
4. E. P. Armendariz, A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc. 18 (1974) 470–473. Crossref, Google Scholar
5. M. Azimi and A. Moussavi, Nilpotent elements in skew polynomial rings, preprint. Google Scholar
6. H. E. Bell, Near-rings in which each element is a power of itself, Bull. Austral. Math. Soc. 2 (1970) 363–368. Crossref, Google Scholar
7. W. Chen, On Nil-semicommutative ings, Thai J. Math. 9 (2011) 39–47. Web of Science, Google Scholar
8. E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 107(3) (2005) 207–224. Crossref, Web of Science, Google Scholar
9. I. N. Herstin and L. W. Small, Nil rings satisfying certain chain conditions, Canad. J. Math. 16 (1964) 771–776. Crossref, Web of Science, Google Scholar
10. S. Hizem, A note on nil power serieswise Armendariz rings, Rend. Cir. Mat. Palermo 59 (2010) 87–99. Crossref, Google Scholar
11. C. Huh, C. O. Kim, E. J. Kim, H. K. Kim and Y. Lee, Nil radicals of power series rings, J. Korian Math. Soc. 42 (2005) 1003–1015. Crossref, Web of Science, Google Scholar
12. C. Huh and C. Ik Lee, On $π$ $π$ -Armendariz rings, Bull. Korean Math. Soc. 44 (2007) 641–649. Crossref, Web of Science, Google Scholar
13. M. Habibi and A. Moussavi, On nil skew Armendariz rings, Asian-European J. Math. 1 (2010) 1–16. Google Scholar
14. C. Y. Hong, T. K. Kwak and S. T. Rizvi, Extensions of generalized Armendariz rings, Algebra Colloq. 13 (2006) 253–266. Link, Web of Science, Google Scholar
15. J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996) 289–300. Google Scholar
16. T. Y. Lam, A. Leroy and J. Matczuk, Primeness, semiprimeness and prime radical of Ore extensions, Comm. Algebra 25 (1997) 2459–2506. Crossref, Web of Science, Google Scholar
17. T. Y. Lam, A First Course in Noncommutative Rings, Springer Lecture Notes in Physics (Springer-Verlag, 1991). Crossref, Google Scholar
18. C. Lanski, Nil subrings of Goldie rings are nilpotent, Canad. J. Math. 21 (1969) 904–907. Crossref, Web of Science, Google Scholar
19. T. H. Lenagan, Nil ideals in rings with finite Krull dimension, J. Algebra 29 (1974) 77–87. Crossref, Web of Science, Google Scholar
20. Z. Liu and R. Zhao, On weak Armendariz rings, Comm. Algebra 34 (2006) 2607–2616. Crossref, Web of Science, Google Scholar
21. G. Marks, Direct product and power series formations over $2$ $2$ -primal rings, in Advances in Ring Theory (Birkhuser, Boston, 1997) pp. 239–245. Crossref, Google Scholar
22. G. Marks, On 2-primal ore extensions, Comm. Algebra 29 (2001) 2113–20123. Crossref, Web of Science, Google Scholar
23. L. Ouyang, Extention of nilpotent p.p. ring, Bull. Iranian Math. Soc. 36 (2010), 169–184. Web of Science, Google Scholar
24. L. Ouyang and J. Liu, On weak $(α, δ)$ $(α, δ)$ -compatible rings, Int. J. Algebra 5 (2011) 1283–1296. Google Scholar
25. M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997) 14–17. Crossref, Web of Science, Google Scholar
26. A. Smoktunowicz, Polynomial rings over nil rings need not be nil, J. Algebra 233 (2000) 427–436. Crossref, Web of Science, Google Scholar