On annihilator ideals in skew polynomial rings

Document Type : Research Paper


Department of Pure Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎Tarbiat Modares University‎, ‎P.O‎. ‎Box 14115-134‎, ‎Tehran‎, ‎Iran.


This article examines annihilators in the skew polynomial ring $R[x;alpha,delta]$. A ring is strongly right $AB$ if every
non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property ($A$) and the conditions asked by P.P. Nielsen. We assume that $R$ is an ($alpha$,$delta$)-compatible ring, and prove that, if $R$ is nil-reversible then the skew polynomial ring $R[x;alpha,delta]$
is strongly right $AB$. It is also shown that, every right duo ring with an automorphism $alpha$ is skew McCoy. Moreover, if $R$ is strongly right $AB$ and skew McCoy, then $R[x;alpha]$ and $R[x;delta]$ have right Property ($A$).


Main Subjects

D.D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265--2272.
E.P. Armendariz, A note on extensions of Baer and p. p. rings, J. Aust. Math. Soc. 18 (1947) 470--473.
F. Azarpanah, O.A.S. Karamzadeh and A. Rezai Aliabad, On ideals consisting entirely of zero divisors, Comm. Algebra 28 (2000) 1061--1073.
J.A. Beachy and W.D. Blair, Rings whose faithful left ideals are cofaithful, Pacific J. Math. 58 (1975), no. 1, 1--13.
H.E. Bell, Near-rings in which each element is a power of itself, Bull. Aust. Math. Soc. 2 (1970) 363--368.
 G.F. Birkenmeier and R.P. Tucci, Homomorphic images and the singular ideal of a strongly right bounded ring, Comm. Algebra 16 (1988), no. 12, 1099--1122.
V. Camillo and P.P. Nielsen, McCoy rings and zero-divisors, J. Pure Appl. Algebra 212 (2008), no. 3, 599--615.
J. Clark, Y. Hirano, H.K. Kim and Y. Lee, On a generalized finite intersection property, Comm. Algebra 40 (2012), no. 6, 2151--2160.
P.M. Cohn, Reversible rings, Bull. Lond. Math. Soc. 31 (1999) 641--648.
W. Cortes, Skew polynomial extensions over zip rings, Inter. J. Math. Sci. 2008 (2008) Article ID 496720, 9 pages.
M.P. Darzin, Rings with central idempotent or nilpotent elements, Proc. Edinb. Math. Soc. 9 (1958), no. 2, 157--165.
C. Faith, Algebra II, Ring Theory, Springer-Verlag, Berlin, 1976.
C. Faith, Commutative FPF rings arising as split-null extensions, Proc. Amer. Math. Soc. 90 (1984) 181--185.
C. Faith, Rings with zero intersection property on annihilator: zip rings, Publ. Math. 33 (1989), no. 2, 329--338.
E.H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc. 89 (1958) 79--91.
M. Habibi, A. Moussavi and A. Alhevaz, The McCoy condition on Ore extensions, Comm. Algebra 41 (2013) 124--141.
E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 3 (2005) 207--224.
M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965) 110--130.
G. Hinkle and J.A. Huckaba, The generalized Kronecker function ring and the ring R(X), J. Reine Angew. Math. 292 (1977) 25--36.
Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168 (2002) 45--52.
C.Y. Hong, N.K. Kim and Y. Lee, Extensions of McCoy's theorem, Glasg Math. J. 52 (2010) 155--159.
C.Y. Hong, N.K. Kim, Y. Lee and S.J. Ryu, Rings with Property (A) and their extensions, J. Algebra 315 (2007) 612--628.
J.A. Huckaba, Commutative rings with zero divisors, Marcel Dekker Inc. New York, 1988.
J.A. Huckaba and J.M. Keller, Annihilator of ideals in commutative rings, Pacific J. Math. 83 (1979), no. 2, 375--379.
C. Huh, Y. Lee and A. Smoktunowicz, Armendariz rings and semi-commutative rings, Comm. Algebra 30 (2002) 751--761.
S.U. Hwang, N.K. Kim and Y. Lee, On rings whose right annihilator are bounded, Glasg Math. J. 51 (2009) 539--559.
N. Jacobson, The Theory of Rings, Amer. Math. Soc. Providence, RI, 1943.
I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
D. Khurana, G. Marks and K. Srivastava, On unit-central rings, Advances in ring theory, 205--212, Trends Math., Birkauser/Springer Basel AG, Basel, 2010.
N. Kim, T.K. Kwak and Y. Lee, Insertion-of-factors-property skewed by ring endomorphisms, Taiwanese J. Math. 18 (2014), no. 3, 849--869.
N. Kim and Y. Lee, Extension of reversible rings, J. Pure Appl. Algebra 185 (2003), no. 1-3, 207--223.
J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289--300.
T.Y. Lam, A First Course in Noncommutative Rings, Grad. Texts in Math. 131, Springer-Verlag, New York, 1991.
T.Y. Lam, A. Leroy and J. Matczuk, Primeness, semiprimeness and prime radical of Ore extensions, Comm. Algebra 25 (1997), no. 8, 2459--2506.
T.K. Lee and Y. Zhou, A unified approach to the Armendariz property of polynomial rings and power series rings, Colloq. Math. 113 (2008), no. 1, 151--169.
T.G. Lucas, Two annihilator conditions: Property (A) and (a.c.), Comm. Algebra 14 (1986), no. 3, 557--580.
G. Marks, Reversible and symmetric rings, J. Pure Appl. Algebra 174 (2002), no. 3, 311--318.
G. Marks, A taxonomy of 2-primal rings, J. Algebra 266 (2003), no. 2, 494--520. 
G. Marks, Duo rings and Ore extensions, J. Algebra 280 (2004) 463--471.
A. Moussavi and E. Hashemi, On (α;δ)-skew Armendariz rings, J. Korean Math. Soci. 42 (2005), no. 2, 353--363.
L.M. de Narbonne, Anneaux semi-commutatifs et unis riels anneaux dont les id aux principaux sont idempotents, in: Proceedings of the 106th National Congress of Learned Societies, pp. 71--73, Bibliotheque Nationale, Paris, 1982.
A.R. Nasr-Isfahani and A. Moussavi, On weakly rigid rings, Glasg Math. J. 51 (2009), no. 3, 425--440.
A.R. Nasr-Isfahani and A. Moussavi, On a quotient of polynomial rings, Comm. Algebra 38 (2010), no. 2, 567--575.
P.P. Nielsen, Semi-commutativity and the McCoy condition, J. Algebra 298 (2006), no. 1, 134--141.
L. Ouyang and G.F. Birkenmeier, Weak annihilator over extension rings, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), no. 2, 345--357.
Y. Quentel, Sur la compacitfi  du spectre minimal d'un anneau, Bull. Soc. Math. France 99 (1971) 265--272.
M.B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14--17.
W. Xue, On strongly right bounded finite rings, Bull. Aust. Math. Soc. 44 (1991), no. 3, 353--355.
W. Xue, Structure of minimal noncommutative duo rings and minimal strongly bounded non-duo rings, Comm. Algebra 20 (1992), no. 9, 2777--2788.
S. Yang, X. Song and Z. Liu, Power-serieswise McCoy rings, Algebra Colloq. 18 (2011), no. 2, 301--310.
J.M. Zelmanowitz, The finite intersection property on annihilator right ideals, Proc. Amer. Math. Soc. 57 (1976), no. 2, 213--216.