QUASI-ARMENDARIZ PROPERTY FOR SKEW POLYNOMIAL RINGS

Title & Authors
QUASI-ARMENDARIZ PROPERTY FOR SKEW POLYNOMIAL RINGS
Baser, Muhittin; Kwa, Tai Keun;

Abstract
The concept of the quasi-Armendariz property of rings properly contains Armendariz rings and semiprime rings. In this paper, we extend the quasi-Armendariz property for a polynomial ring to the skew polynomial ring, hence we call such ring a $\small{{\sigma}}$-quasi-Armendariz ring for a ring endomorphism $\small{{\sigma}}$, and investigate its structures, several extensions and related properties. In particular, we study the semiprimeness and the quasi-Armendariz property between a ring R and the skew polynomial ring R[x;$\small{{\sigma}}$$] of R, and so these provide us with an opportunity to study quasi-Armendariz rings and semiprime rings in a general setting, and several known results follow as consequences of our results. Keywords quasi-Armendariz property;skew polynomial ring;semiprime ring;rigid ring; Language English Cited by 1. Quasi-Armendariz generalized power series rings, Journal of Algebra and Its Applications, 2016, 15, 05, 1650086 References 1. E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470-473. 2. M. Baser, A. Harmanci, and T. K. Kwak, Generalized semicommutative rings and their extensions, Bull. Korean Math. Soc. 45 (2008), no. 2, 285-297. 3. M. Baser, F. Kaynarca, T. K. Kwak, and Y. Lee, Weak quasi-Armendariz rings, to apperar in Algebra Colloq. 4. W. Cortes, Skew Armendariz rings and annihilator ideals of skew polynomial rings, Algebraic structures and their representations, 249-259, Contemp. Math., 376, Amer. Math. Soc., Providence, RI, 2005. 5. Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168 (2002), no. 1, 45-52. 6. C. Y. Hong, N. K. Kim, and T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra 151 (2000), no. 3, 215-226. 7. C. Y. Hong, N. K. Kim, and T. K. Kwak, On skew Armendariz rings, Comm. Algebra 31 (2003), no. 1, 103-122. 8. C. Y. Hong, N. K. Kim, and T. K. Kwak, On quasi-rigid ideals and rings, Bull. Korean Math. Soc. 47 (2010), no. 2, 385-399. 9. C. Y. Hong, N. K. Kim, and Y. Lee, Skew polynomial rings over semiprime rings, J. Korean Math. Soc. 47 (2010), no. 5, 879-897. 10. C. Y. Hong, T. K. Kwak, and S. T. Rizvi, Extensions of generalized Armendariz rings, Algebra Colloq. 13 (2006), no. 2, 253-266. 11. A. A. M. Kamal, Some remarks on Ore extension rings, Comm. Algebra 22 (1994), no. 10, 3637-3667. 12. N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488. 13. J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300. 14. T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991. 15. T. K. Lee and Y. Q. Zhou, Armendariz and reduced rings, Comm. Algebra 32 (2004), no. 6, 2287-2299. 16. J. Matczuk, A characterization of$\sigma\$-rigid rings, Comm. Algebra 32 (2004), no. 11, 4333-4336.

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