| Title:
|
Left APP-property of formal power series rings (English) |
| Author:
|
Liu, Zhongkui |
| Author:
|
Yang, Xiaoyan |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
44 |
| Issue:
|
3 |
| Year:
|
2008 |
| Pages:
|
185-189 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A ring $R$ is called a left APP-ring if the left annihilator $l_R(Ra)$ is right $s$-unital as an ideal of $R$ for any element $a\in R$. We consider left APP-property of the skew formal power series ring $R[[x; \alpha ]]$ where $\alpha $ is a ring automorphism of $R$. It is shown that if $R$ is a ring satisfying descending chain condition on right annihilators then $R[[x; \alpha ]]$ is left APP if and only if for any sequence $(b_0, b_1, \dots )$ of elements of $R$ the ideal $l_R$ $\big (\sum _{j=0}^{\infty }\sum _{k=0}^{\infty }R\alpha ^k(b_j)\big )$ is right $s$-unital. As an application we give a sufficient condition under which the ring $R[[x]]$ over a left APP-ring $R$ is left APP. (English) |
| Keyword:
|
left APP-ring |
| Keyword:
|
skew power series ring |
| Keyword:
|
left principally quasi-Baer ring |
| MSC:
|
16P60 |
| MSC:
|
16W60 |
| idZBL:
|
Zbl 1203.16031 |
| idMR:
|
MR2462973 |
| . |
| Date available:
|
2009-01-29T09:14:47Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119757 |
| . |
| Reference:
|
[1] Birkenmeier, G. F., Kim, J. Y., Park, J. K.: A sheaf representation of quasi-Baer rings.J. Pure Appl. Algebra 146 (2000), 209–223. Zbl 0947.16018, MR 1742340, 10.1016/S0022-4049(99)00164-4 |
| Reference:
|
[2] Birkenmeier, G. F., Kim, J. Y., Park, J. K.: On polynomial extensions of principally quasi-Baer rings.Kyungpook Math. J. 40 (2000), 247–254. Zbl 0987.16017, MR 1803098 |
| Reference:
|
[3] Birkenmeier, G. F., Kim, J. Y., Park, J. K.: On quasi-Baer rings.Contemp. Math. 259 (2000), 67–92. Zbl 0974.16006, MR 1778495, 10.1090/conm/259/04088 |
| Reference:
|
[4] Birkenmeier, G. F., Kim, J. Y., Park, J. K.: Principally quasi-Baer rings.Comm. Algebra 29 (2001), 639–660. Zbl 0991.16005, MR 1841988, 10.1081/AGB-100001530 |
| Reference:
|
[5] Fraser, J. A., Nicholson, W. K.: Reduced PP-rings.Math. Japon. 34 (1989), 715–725. Zbl 0688.16024, MR 1022149 |
| Reference:
|
[6] Hirano, Y.: On annihilator ideals of a polynomial ring over a noncommutative ring.J. Pure Appl. Algebra 168 (2002), 45–52. Zbl 1007.16020, MR 1879930, 10.1016/S0022-4049(01)00053-6 |
| Reference:
|
[7] Liu, Z.: A note on principally quasi-Baer rings.Comm. Algebra 30 (2002), 3885–3890. Zbl 1018.16023, MR 1922317, 10.1081/AGB-120005825 |
| Reference:
|
[8] Liu, Z., Ahsan, J.: PP-rings of generalized power series.Acta Math. Sinica 16 (2000), 573–578, English Series. Zbl 1015.16046, MR 1813453 |
| Reference:
|
[9] Liu, Z., Zhao, R.: A generalization of PP-rings and p.q.-Baer rings.Glasgow Math. J. 48 (2006), 217–229. Zbl 1110.16003, MR 2256973, 10.1017/S0017089506003016 |
| Reference:
|
[10] Stenström, B.: Rings of Quotients.Springer-Verlag, Berlin, 1975. MR 0389953 |
| Reference:
|
[11] Tominaga, H.: On $s$-unital rings.Math. J. Okayama Univ. 18 (1976), 117–134. Zbl 0335.16020, MR 0419511 |
| . |
