# ON ANNIHILATOR IDEALS OF A NEARRING OF SKEW POLYNOMIALS OVER A RING

• Hashemi, Ebrahim
• Published : 2007.11.30
• 74 12

#### Abstract

For a ring endomorphism ${\alpha}$ and an ${\alpha}-derivation\;{\delta}$ of a ring R, we study relation between the set of annihilators in R and the set of annihilators in nearring $R[x;{\alpha},{\delta}]\;and\;R_0[[x;{\alpha}]]$. Also we extend results of Armendariz on the Baer and p.p. conditions in a polynomial ring to certain analogous annihilator conditions in a nearring of skew polynomials. These results are somewhat surprising since, in contrast to the skew polynomial ring and skew power series case, the nearring of skew polynomials and skew power series have substitution for its "multiplication" operation.

#### Keywords

annihilator conditions;nearrings;skew power series;skew polynomial rings;Baer ring;quasi-Baer rings;${\alpha}-rigid$ rings;Rickart rings

#### References

1. D. D. Anderson and V. Camillo, Armendariz rings and Gaussian rings, Comm. Algebra 26 (1998), no. 7, 2265-2275 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. S. K. Berberian , Baer $\ast-rings$, Springer-Verlag, Berlin, 1972
4. G. F. Birkenmeier and F. K. Huang, Annihilator conditions on polynomials, Comm.Algebra 29 (2001), no. 5, 2097-2112 https://doi.org/10.1081/AGB-100002172
5. G. F. Birkenmeier and F. K. Huang, Annihilator conditions on formal power series, Algebra colloq. 9 (2002), no. 1, 29-37
6. G. F. Birkenmeier and F. K. Huang, Annihilator conditions on polynomials II, Monatsh. Math. 141 (2004), no. 4, 265-276 https://doi.org/10.1007/s00605-003-0056-z
7. G. F. Birkenmeier, J. Y. Kim, and J. K. Park, Quasi-Baer ring extensions and biregular rings, Bull. Austral. Math. Soc. 61 (2000), no. 1, 39-52 https://doi.org/10.1017/S0004972700022000
8. G. F. Birkenmeier, Polynomial extensions of Baer and quasi-Baer rings, J. Pure Appl. Algebra 159 (2001), no. 1, 25-42 https://doi.org/10.1016/S0022-4049(00)00055-4
9. G. F. Birkenmeier, Principally quasi-Baer rings, Comm. Algebra 29 (2001), no. 2, 639-660 https://doi.org/10.1081/AGB-100001530
10. W. E. Clark, Twisted matrix units semigroup algebra, Duke Math. J. 34 (1967), 417- 423 https://doi.org/10.1215/S0012-7094-67-03446-1
11. J. A. Fraser and W. K. Nicholson, Reduced PP-rings, Math. Japon. 34 (1989), no. 5, 715-725
12. E. Hashemi and A. Moussavi, Polynomial extensions of quasi-Baer rings, Acta Math. Hungar. 107 (2005), no. 3, 207-224 https://doi.org/10.1007/s10474-005-0191-1
13. E. Hashemi and A. Moussavi, Skew power series extensions of $\alpha-rigid$ p.p.-rings, Bull. Korean Math. Soc. 41 (2004), no. 4, 657-665 https://doi.org/10.4134/BKMS.2004.41.4.657
14. 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
15. C. Y. Hong , N. K. Kim, and T. K. Kwak, Ore extensions of Baer and p.p.-rings, J. Pure Appl. Algebra 151 (2000), 215-226 https://doi.org/10.1016/S0022-4049(99)00020-1
16. I. Kaplansky, Rings of operators, Benjamin, New York, 1968
17. J. Krempa, Some examples of reduced rings, Algebra Colloq. 3 (1996), no. 4, 289-300
18. A. Moussavi and E. Hashemi, On $(\alpha,\;\delta)$ Skew Armendariz rings, J. Korean Math. Soc. 42 (2005), no. 2, 353-363 https://doi.org/10.4134/JKMS.2005.42.2.353