Korean J. Math. Vol. 21 No. 4 (2013) pp.421-428
DOI: https://doi.org/10.11568/kjm.2013.21.4.421

On coefficients of nilpotent polynomials in skew polynomial rings

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Sang Bok Nam
Sung Ju Ryu
Sang Jo Yun


We observe the basic structure of the products of coefficients of nilpotent (left) polynomials in skew polynomial rings. This study consists of a process to extend a well-known result for semi-Armendariz rings. We introduce the concept of {\it $\alpha$-skew $n$-semi-Armendariz ring}, where $\alpha$ is a ring endomorphism. We prove that a ring $R$ is $\alpha$-rigid if and only if the $n$ by $n$ upper triangular matrix ring over $R$ is $\bar\alpha$-skew $n$-semi-Armendariz. This result are applicable to several known results.

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[1] D.D. Anderson, V. Camillo, Armendariz rings and Gaussian rings, Comm. Al- gebra 26 (1998), 2265–2272. Google Scholar

[2] D.D. Anderson, V. Camillo, Semigroups and rings whose zero products commute, Comm. Algebra 27 (1999), 2847–2852. Google Scholar

[3] E.P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470–473. Google Scholar

[4] Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168, (2002), 45–52. Google Scholar

[5] C.Y. Hong, N.K. Kim, T.K. Kwak, On skew Armendariz rings, Comm. Algebra 31 (2003), 103–122. Google Scholar

[6] C.Y. Hong, N.K. Kim, T.K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure and Appl. Algebra 151 (2000), 215–226. Google Scholar

[7] C. Huh, Y. Lee, A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), 751–761. Google Scholar

[8] Y.C. Jeon, Y. Lee, S.J. Ryu, A structure on coefficients of nilpotent polynomials, J. Korean Math. Soc. 47 (2010), 719–733. Google Scholar

[9] N.K. Kim, Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), 477–488. Google Scholar

[10] J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), 289–300. Google Scholar

[11] J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359-368. Google Scholar

[12] T.-K. Lee, T.-L. Wong, On Armendariz rings, Houston J. Math. 2 (2003), 583–593. Google Scholar

[13] M.B. Rege, S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), 14–17. Google Scholar