On skew power series over McCoy rings

Document Type : Research Paper

Authors

Department of Mathematics, Higher Education center of Eghlid, Eghlid, Iran

Abstract

Let $R$ be a ring with an endomorphism $\alpha$‎. ‎A ring $R$ is a skew power series McCoy ring if whenever any non-zero power series $f(x)=\sum_{i=0}^{\infty}a_ix^i,g(x)=\sum_{j=0}^{\infty}b_jx^j\in R[[x;\alpha]]$ satisfy $f(x)g(x)=0$‎, ‎then there exists a non-zero element $c\in R$ such that $a_ic=0$‎, ‎for all $i=0,1,\ldots$‎. ‎We investigate relations between the skew power series ring and the standard ring-theoretic properties‎. ‎Moreover‎, ‎we obtain some characterizations for skew power series ring $R[[x;\alpha]]$‎, ‎to be McCoy‎, ‎zip‎, ‎strongly \textit{AB} and has Property (A)‎.

Keywords

References

[1] A. Alhevaz and E. Hashemi , An alternative perspective on skew generalized power series ring, Mediterr. J. Math.13 (2016), 4723{4744.
[2] A. Alhevaz, E. Hashemi and R. Mohammad, On transfer of annihilator conditions of rings, J. Algebra Appl 17, No 10(2018), 185{199.
[3] A. Alhevaz and D. Kiani, McCoy property of skew Laurent polynomials and power series rings, J. Algebra Appl 13(2)(2014), 1350083 (23 pages).
[4] V. Camillo and P.P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra., 212, No 3 (2008), 599{615.
[5] D.E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc., 27 (1971), 427{433.
[6] R. Gilmer, A. Grams and T. Parker, Zero divisors in power series rings, J. Reine Angew. Math., 278/279 (1975), 145{164.
[7] E. Hashemi, A. As. Estaji and M. Ziembowski, Answers to some questions concerning rings with Property (A) Proc. Edinb. Math. Soc., in prees.
[8] T.K. Kwak and Y. Lee, Zero divisors in skew power series rings, Comm. Algebra., 43, No 10(2015), 4427{4445.
[9] T.Y. Lam, A First Course in Noncommutative Rings, Second edition. Graduate Texts in Mathematics, 131. Springer-Verlag, New York, 2001.

Files

History

  • Receive Date: 28 March 2022
  • Revise Date: 30 May 2022
  • Accept Date: 03 August 2022

Share

How to cite

Statistics