Rings with the annihilator condition and their extensions

  • Latifa Malki
  • Mohamed Louzari
Keywords: Annihilator condition, Annihilators, p.q.-Baer rings, α-skew quasi Armendariz rings.


Let R be an associative unital ring and let α be an endomorphism of R. In this article, we study rings having the annihilator condition (a.c.) and some related rings. Also, many polynomial extensions of R having the condition (a.c.) are stud-ied. In particular, α-skew quasi-Armendariz rings and α-(sps) quasi-Armendariz rings. We prove that this class of rings have always the property (a.c.) on the left but not necessary on the right. Furthermore, we investigate the transmission of the property (a.c) from a ring R to R[x, α] and R[[x, α]].


[1] Henriksen, M., Jerison, M. (1965). The space of minimal prime ideals of a commutative ring. Trans. Amer.
Math. Soc. 115:110-130. DOI: 10.1090/S0002-9947-1965-0194880-9
[2] Lucas, T. G. (1986). Two annihilator conditions: Property (A) and (a.c.). Comm. Algebra 14(3):557-580.
DOI: 10.1080/00927878608823325
[3] Huckaba, J. A., Keller, J. M. (1979). Annihilation of ideals in commutative rings. Pacif i c J. Math.
83(2):375-379. DOI: 10.2140/pjm.1979.83.375
[4] Bajor, G., Ziembowski, M. (2019). Annihilator condition does not pass to polynomials and power series.
J. Pure Appl. Algebra 223(9):3869-3878. DOI: 10.1016/j.jpaa.2018.12.009
[5] Hong, C. Y., Kim, N. K., Nielsen, P. P., Lee, Y. (2009). The minimal prime spectrum of rings with annihilator conditions. J. Pure Appl. Algebra 213(7):1478-1488. DOI: 10.1016/j.jpaa.2009.01.005
[6] Armendariz, E. P. (1974). A note on extensions of Baer an P.P. rings. J. Austral. Math, Soc. 18(4):470-473.
DOI: 10.1017/S1446788700029190
[7] Hirano, Y. (2002). On annihilator ideals of a polynomial ring over a noncommutative ring. J. Pure Appl.
Algebra. 168(1):45-52. DOI: 10.1016/S0022-4049(01)00053-6
[8] Hong, C. Y., Kim, N. K., Kwak, T. K. (2003). On skew Armendariz rings. Comm. Algebra. 31(1):103-122. DOI: 10.1081/AGB-120016752
[9] Rege, M. B., Chhawchharia, S. (1997). Armendariz rings. Proc. Japan. Acad. Ser. A Math. Sci. 73(1):14-17. DOI: 10.3792/pjaa.73.14
[10] Baser, M., Kwak, T. K. (2011). Quasi-Armendariz property for skew polynomial rings. Comm. Korean Math. Soc. 26(4):557-573. DOI: 10.4134/CKMS.2011.26.4.557
[11] Törner, G., Mazurek, R. (2004). Comparizer ideals of rings. Comm. Algebra. 32(12):4653-4665. DOI:
[12] Harmanci, A., Baser, M., Kwak, T. K. (2008). Generalized semicommutative rings and their extensions.
Bull. Korean Math. Soc. 45(2):285-297. DOI:10.4134/BKMS.2008.45.2.285
[13] Clark, W. E. (1967) Twisted matrix units semigroup algebras. Duke Math. J., 34(3):417-424. DOI:
[14] Birkenmeier, G. F.,Park, J. K., Kim, J. Y. (2001).Principally quasi-Baer rings. Comm. Algebra. 29(2):639-660. DOI: 10.1081/AGB-100001530
[15] Taherifar, A., Dube, T. (2021). On the lattice of annihilator ideals and its applications. Comm. Algebra 49(6):2444-2456. DOI:10.1080/00927872.2021.1872588
[16] Hong, C. Y., Kim, N. K., Lee, Y. (2010). Skew polynomial rings over semiprime rings. J. Korean Math.
Soc. 47(5):879-897. DOI: 10.4134/JKMS.2010.47.5.879
[17] Zaks, A., Pollingher, P. (1970). On Baer and quasi-Baer rings. Duke Math. J. 37(1):127-138. DOI:
[18] Birkenmeier, G. F., Park, J. K., Rizvi, S. T. (2013) Extensions of Rings and Modules. New York, USA:
[19] Ben Yakoub, L., Louzari, M. (2009) Ore extensions of principally quasi-Baer rings. JP Journal of Algebra, Number Theory and Appl. 13(2):137-151. DOI: 10.48550/arXiv.0709.0325
[20] Moussavi, A., Paykan, K. (2016) Quasi-Armendariz generalized power series rings. J. Algebra Appl.
15(5):1650086. DOI: 10.1142/S0219498816500869
[21] Birkenmeier, G. F., Park, J. K., Kim, J. Y. (2000) On polynomial extensions of principally quasi-Baer rings.
Kyungpook Math. J. 40(2):247-253.
[22] Hong, C. Y., Kim, N. K., Kwak, T. K. (2000). Ore extensions of Baer and p.p.-rings. J. Pure Appl.
Algebra 151(3):215-226. DOI: 10.1016/S0022-4049(99)00020-1
[23] Krempa, J. (1996) Some examples of reduced rings. Algebra Colloq. 3(4):289-300.
[24] Yohe, C. R. (1968). On rings in which every ideal is the annihilator of an element. Proc. Amer. Math. Soc.