Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Armendariz ring; $(\alpha ,\delta )$-skew Armendariz ring; weak Armendariz ring; weak $(\alpha ,\delta )$-skew Armendariz ring
Armendariz ring; $(\alpha ,\delta )$-skew Armendariz ring; weak Armendariz ring; weak $(\alpha ,\delta )$-skew Armendariz ring
Summary:
In this note, for a ring endomorphism $\alpha $ and an $\alpha $-derivation $\delta $ of a ring $R$, the notion of weakened $(\alpha ,\delta )$-skew Armendariz rings is introduced as a generalization of $\alpha $-rigid rings and weak Armendariz rings. It is proved that $R$ is a weakened $(\alpha ,\delta )$-skew Armendariz ring if and only if $T_n(R)$ is weakened $(\bar {\alpha },\bar {\delta })$-skew Armendariz if and only if $R[x]/(x^n)$ is weakened $(\bar {\alpha },\bar {\delta })$-skew Armendariz ring for any positive integer $n$.
References:
[1] Alhevaz, A., Moussavi, A., Habibi, M.: On rings having McCoy-like conditions. Commun. Algebra 40 (2012), 1195-1221. DOI 10.1080/00927872.2010.548842 | MR 2912979 | Zbl 1260.16024
[2] Anderson, D. D., Camillo, V.: Armendariz rings and Gaussian rings. Commun. Algebra 26 (1998), 2265-2272. DOI 10.1080/00927879808826274 | MR 1626606 | Zbl 0915.13001
[3] Anderson, D. D., Camillo, V.: Semigroups and rings whose zero products commute. Commun. Algebra 27 (1999), 2847-2852. DOI 10.1080/00927879908826596 | MR 1687281 | Zbl 0929.16032
[4] Annin, S.: Associated primes over skew polynomial rings. Commun. Algebra 30 (2002), 2511-2528. DOI 10.1081/AGB-120003481 | MR 1904650 | Zbl 1010.16025
[5] Armendariz, E. P.: A note on extensions of Baer and P.P.-rings. J. Aust. Math. Soc. 18 (1974), 470-473. DOI 10.1017/S1446788700029190 | MR 0366979 | Zbl 0292.16009
[6] Chen, J., Yang, X., Zhou, Y.: On strongly clean matrix and triangular matrix rings. Commun. Algebra 34 (2006), 3659-3674. DOI 10.1080/00927870600860791 | MR 2262376 | Zbl 1114.16024
[7] Chen, J., Zhou, Y.: Extensions of injectivity and coherent rings. Commun. Algebra 34 (2006), 275-288. DOI 10.1080/00927870500346263 | MR 2194766 | Zbl 1112.16002
[8] Chen, W., Cui, S.: On weakly semicommutative rings. Commun. Math. Res. 27 (2011), 179-192. MR 2808276 | Zbl 1249.16041
[9] Habibi, M., Moussavi, A.: On nil skew Armendariz rings. Asian-Eur. J. Math. 5 (2012), Article ID 1250017, 16 pages. DOI 10.1142/S1793557112500179 | MR 2959844 | Zbl 1263.16028
[10] Hashemi, E., Moussavi, A.: Polynomial extensions of quasi-Baer rings. Acta. Math. Hung. 107 (2005), 207-224. DOI 10.1007/s10474-005-0191-1 | MR 2148584 | Zbl 1081.16032
[11] Hirano, Y.: On the uniqueness of rings of coefficients in skew polynomial rings. Publ. Math. 54 (1999), 489-495. MR 1694531 | Zbl 0930.16018
[12] Hong, C. Y., Kim, H. K., Kim, N. K., Kwak, T. K., Lee, Y., Park, K. S.: Rings whose nilpotent elements form a Levitzki radical ring. Commun. Algebra 35 (2007), 1379-1390. DOI 10.1080/00927870601117597 | MR 2313674 | Zbl 1121.16021
[13] Hong, C. Y., Kim, N. K., Kwak, T. K.: Ore extensions of Baer and p.p.-rings. J. Pure Appl. Algebra 151 (2000), 215-226. DOI 10.1016/S0022-4049(99)00020-1 | MR 1776431 | Zbl 0982.16021
[14] Huh, C., Lee, Y., Smoktunowicz, A.: Armendariz rings and semicommutative rings. Commun. Algebra 30 (2002), 751-761. DOI 10.1081/AGB-120013179 | MR 1883022 | Zbl 1023.16005
[15] Kim, N. K., Lee, Y.: Armendariz rings and reduced rings. J. Algebra 223 (2000), 477-488. DOI 10.1006/jabr.1999.8017 | MR 1735157 | Zbl 0957.16018
[16] Kim, N. K., Lee, Y.: Extensions of reversible rings. J. Pure Appl. Algebra 185 (2003), 207-223. DOI 10.1016/S0022-4049(03)00109-9 | MR 2006427 | Zbl 1040.16021
[17] Krempa, J.: Some examples of reduced rings. Algebra Colloq. 3 (1996), 289-300. MR 1422968 | Zbl 0859.16019
[18] Lam, T. Y., Leroy, A., Matczuk, J.: Primeness, semiprimeness and prime radical of Ore extensions. Commun. Algebra 25 (1997), 2459-2506. DOI 10.1080/00927879708826000 | MR 1459571 | Zbl 0879.16016
[19] Lambek, J.: On the representation of modules by sheaves of factor modules. Can. Math. Bull. 14 (1971), 359-368. DOI 10.4153/CMB-1971-065-1 | MR 0313324 | Zbl 0217.34005
[20] Liu, Z., Zhao, R.: On weak Armendariz rings. Commun. Algebra 34 (2006), 2607-2616. DOI 10.1080/00927870600651398 | MR 2240395 | Zbl 1110.16026
[21] Moussavi, A., Hashemi, E.: On ($\alpha,\delta$)-skew Armendariz rings. J. Korean Math. Soc. 42 (2005), 353-363. DOI 10.4134/JKMS.2005.42.2.353 | MR 2121504 | Zbl 1090.16012
[22] Rege, M. B., Chhawchharia, S.: Armendariz rings. Proc. Japan Acad., Ser. A 73 (1997), 14-17. DOI 10.3792/pjaa.73.14 | MR 1442245 | Zbl 0960.16038
[23] Shin, G.: Prime ideals and sheaf representation of a pseudo symmetric rings. Trans. Am. Math. Soc. 184 (1973), 43-60. DOI 10.1090/S0002-9947-1973-0338058-9 | MR 0338058 | Zbl 0283.16021
[24] Wang, Y., Jiang, M., Ren, Y.: Ore extensions over weakly 2-primal rings. Commun. Math. Res. 32 (2016), 70-82. DOI 10.13447/j.1674-5647.2016.01.05 | MR 3467810 | Zbl 1363.16075
Search
Partner of
