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Journals / Turkish Journal of Mathematics / 2008 / Cilt: 32 - Sayı: 4

Rickart-type Annihilator Conditions on Formal Power Series

Ebrahim HASHEMI
Pages
363–372
Publication Date
2022-11-05
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

An example to show that, a-rigid condition on R is not superfluous, is provided.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Let R be an a-rigid ring and R0[[x;a]] be the nearring of a formal skew power series in which addition and substitution are used as operations.
📊
Key Findings & Data:It is shown that R is Rickart and any countable family of idempotents of R has a join in I(R) if and only if R0[[x;a]]\in Rr1 if and only if R0[[x;a]]\in R\ell 1 if and only if R0[[x;a]]\in qRr2.
💡
Conclusion & Impact:An example to show that, a-rigid condition on R is not superfluous, is provided.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Annihilator conditions · Nearrings · Skew power series

Abstract

Let R be an a-rigid ring and R0[[x;a]] be the nearring of a formal skew power series in which addition and substitution are used as operations. It is shown that R is Rickart and any countable family of idempotents of R has a join in I(R) if and only if R0[[x;a]]\in Rr1 if and only if R0[[x;a]]\in R\ell 1 if and only if R0[[x;a]]\in qRr2. An example to show that, a-rigid condition on R is not superfluous, is provided.

Özet

Let R be an a-rigid ring and R0[[x;a]] be the nearring of a formal skew power series in which addition and substitution are used as operations. It is shown that R is Rickart and any countable family of idempotents of R has a join in I(R) if and only if R0[[x;a]]\in Rr1 if and only if R0[[x;a]]\in R\ell 1 if and only if R0[[x;a]]\in qRr2. An example to show that, a-rigid condition on R is not superfluous, is provided.

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Cite This Article

HASHEMI, E. (2022). Rickart-type Annihilator Conditions on Formal Power Series. Turkish Journal of Mathematics, Cilt: 32 - Sayı: 4, 363–372. https://ajindex.com/articles/609048

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References (23)

  1. Anderson, D.D. and Camillo, V.: Armendariz rings and Gaussian rings. Comm. Algebra (7), 2265-2275 (1998).
  2. Armendariz, E.P.: A note on extensions of Baer and p.p.-rings, J. Austral. Math. Soc. 18, 473 (1974).
  3. Berberian, S.K.: Baer*-rings; Springer-Verlag: Berlin, 1968.
  4. Birkenmeier, G.F., Kim, J.Y. and Park, J.K.: Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl. Algebra 159, 24-42 (2001).
  5. Birkenmeier, G.F. and Huang, F.K.: Annihilator conditions on polynomials, Comm. Alge- bra 29(5), 2097-2112 (2001).
  6. Birkenmeier, G.F. and Huang, F.K.: Annihilator conditions on polynomials II, Monatsh. Math. (2003).
  7. Birkenmeier, G.F. and Huang, F.K.: Annihilator conditions on formal power series, Algebra Colloq. 9(1), 29-37 (2002).
  8. Fraser, J.A., Nicholson, W.K.: Reduced PP-rings. Math. Japonica 34(5), 715-725 (1989).
  9. Hashemi, E. and Moussavi, A.: Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 107(3), 207-224 (2005).
  10. Hashemi, E. and Moussavi, A.: Skew power series extensions of α-rigid p.p.-rings, Bull. Korean Math. Soc. 41(4), 657-665 (2004).
  11. Hirano, Y.: On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168, 45-52 (2002).
  12. Hong, C.Y., Kim, N.K. and Kwak, T.K.: Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra 151 (2000) 215-226.
  13. Hong, C.Y., Kim, N.K. and Kwak, T.K.: On skew Armendariz rings, Comm. Algebra 31(1), 122 (2003).
  14. Huh, C., Lee, Y. and Smoktunowicz, A.: Armendariz rings and semicommutative rings, Comm. Algebra 30(2), 751-761 (2002).
  15. Kaplansky, I.: Rings of Operators, Benjamin, New York, 1965.
  16. Kim, N.K. and Lee, Y.: Armendariz rings and reduced rings, J. Algebra 223, 477-488 (2000).
  17. Krempa, J.: Some examples of reduced rings, Algebra Colloq. 3(4), 289-300 (1996).
  18. Lee, T.K. and Wong, T.L.: On Armendariz rings, Houston J. Math. 29(3), 583-593 (2003).
  19. Liu, Z.: A note on principally quasi-Baer rings, Comm. Algebra 30(8), 3885-3890 (2002).
  20. Liu, Z. and Ahsan, J.: PP-rings of generalized power series, Acta Math. Sinica English Series 16, 573-578 (2000).
  21. Matczuk, J.: A characterization of α-rigid rings, Comm. Algebra 32(1), 4333-4337 (2004).
  22. Moussavi, A. and Hashemi, E.: On (α, δ)-skew Armendariz Rings, J. Korean Math. Soc. (2), 353-363 (2005).
  23. Rege, M.B., Chhawchharia, S.: Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73, 17 (1997). Ebrahim HASHEMI Department of Mathematics, Shahrood University of Thechnology, Shahrood-IRAN e-mail: eb [email protected] e-mail: eb hashemi@yaho