ON II-ARMENDARIZ RINGS

• Huh, Chan (DEPARTMENT OF MATHEMATICS BUSAN NATIONAL UNIVERSITY) ;
• Lee, Chang-Ik (DEPARTMENT OF MATHEMATICS BUSAN NATIONAL UNIVERSITY) ;
• Park, Kwang-Sug (DEPARTMENT OF MATHEMATICS BUSAN NATIONAL UNIVERSITY) ;
• Ryu, Sung-Ju (DEPARTMENT OF MATHEMATICS BUSAN NATIONAL UNIVERSITY)
• Published : 2007.11.30
• 78 6

Abstract

We in this note introduce a concept, so called ${\pi}-Armendariz$ ring, that is a generalization of both Armendariz rings and 2-primal rings. We first observe the basic properties of ${\pi}-Armendariz$ rings, constructing typical examples. We next extend the class of ${\pi}-Armendariz$ rings, through various ring extensions.

Keywords

${\pi}-Armendariz$ ring;2-primal ring;Armendariz ring;nilpotent element

References

1. D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2272 https://doi.org/10.1080/00927879808826274
2. E. P. Armendariz, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc. 18 (1974), 470-473 https://doi.org/10.1017/S1446788700029190
3. G. F. Birkenmeier, J. Y. Kim, and J. K. Park, Regularity conditions and the simplicity of prime factor rings, J. Pure Appl. Algebra 115 (1997), no. 3, 213-230 https://doi.org/10.1016/S0022-4049(96)00011-4
4. Y. Hirano, On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra 168 (2002), no. 1, 45-52 https://doi.org/10.1016/S0022-4049(01)00053-6
5. C. Huh, Y. Lee, and A. Smoktunowicz, Armendariz rings and semicommutative rings, Comm. Algebra 30 (2002), no. 2, 751-761 https://doi.org/10.1081/AGB-120013179
6. N. K. Kim and Y. Lee, Armendariz rings and reduced rings, J. Algebra 223 (2000), no. 2, 477-488 https://doi.org/10.1006/jabr.1999.8017
7. T.-K. Lee and T.-L. Wong, On Armendariz rings, Houston J. Math. 29 (2003), no. 3, 583-593
8. G. Marks, A taxonomy of 2-primal rings, J. Algebra 266 (2003), no. 2, 494-520 https://doi.org/10.1016/S0021-8693(03)00301-6
9. M. B. Rege and S. Chhawchharia, Armendariz rings, Proc. Japan Acad. Ser. A Math. Sci. 73 (1997), no. 1, 14-17 https://doi.org/10.3792/pjaa.73.14
10. G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer. Math. Soc. 184 (1973), 43-60 https://doi.org/10.2307/1996398
11. G. F. Birkenmeier, H. E. Heatherly, and E. K. Lee, Completely prime ideals and as- sociated radicals, Ring theory (Granville, OH, 1992), 102-129, World Sci. Publ., River Edge, NJ, 1993
12. G. Marks, On 2-primal Ore extensions, Comm. Algebra 29 (2001), no. 5, 2113-2123 https://doi.org/10.1081/AGB-100002173

Cited by

1. On Rings Having McCoy-Like Conditions vol.40, pp.4, 2012, https://doi.org/10.1080/00927872.2010.548842
2. Nil-Armendariz rings relative to a monoid vol.2, pp.1, 2013, https://doi.org/10.1007/s40065-012-0040-3
3. ON Π-NEAR-ARMENDARIZ RINGS vol.02, pp.01, 2009, https://doi.org/10.1142/S1793557109000078
4. On nilpotent elements of ore extensions vol.10, pp.03, 2017, https://doi.org/10.1142/S1793557117500437
5. On linearly weak Armendariz rings vol.219, pp.4, 2015, https://doi.org/10.1016/j.jpaa.2014.05.039
6. On Skew Triangular Matrix Rings vol.22, pp.02, 2015, https://doi.org/10.1142/S1005386715000243
7. π-Armendariz rings relative to a monoid vol.11, pp.4, 2016, https://doi.org/10.1007/s11464-016-0561-8