Previous |  Up |  Next

Article

Keywords:
left APP-ring; skew power series ring; left principally quasi-Baer ring
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.
References:
[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. DOI 10.1016/S0022-4049(99)00164-4 | MR 1742340 | Zbl 0947.16018
[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. MR 1803098 | Zbl 0987.16017
[3] Birkenmeier, G. F., Kim, J. Y., Park, J. K.: On quasi-Baer rings. Contemp. Math. 259 (2000), 67–92. DOI 10.1090/conm/259/04088 | MR 1778495 | Zbl 0974.16006
[4] Birkenmeier, G. F., Kim, J. Y., Park, J. K.: Principally quasi-Baer rings. Comm. Algebra 29 (2001), 639–660. DOI 10.1081/AGB-100001530 | MR 1841988 | Zbl 0991.16005
[5] Fraser, J. A., Nicholson, W. K.: Reduced PP-rings. Math. Japon. 34 (1989), 715–725. MR 1022149 | Zbl 0688.16024
[6] Hirano, Y.: On annihilator ideals of a polynomial ring over a noncommutative ring. J. Pure Appl. Algebra 168 (2002), 45–52. DOI 10.1016/S0022-4049(01)00053-6 | MR 1879930 | Zbl 1007.16020
[7] Liu, Z.: A note on principally quasi-Baer rings. Comm. Algebra 30 (2002), 3885–3890. DOI 10.1081/AGB-120005825 | MR 1922317 | Zbl 1018.16023
[8] Liu, Z., Ahsan, J.: PP-rings of generalized power series. Acta Math. Sinica 16 (2000), 573–578, English Series. MR 1813453 | Zbl 1015.16046
[9] Liu, Z., Zhao, R.: A generalization of PP-rings and p.q.-Baer rings. Glasgow Math. J. 48 (2006), 217–229. DOI 10.1017/S0017089506003016 | MR 2256973 | Zbl 1110.16003
[10] Stenström, B.: Rings of Quotients. Springer-Verlag, Berlin, 1975. MR 0389953
[11] Tominaga, H.: On $s$-unital rings. Math. J. Okayama Univ. 18 (1976), 117–134. MR 0419511 | Zbl 0335.16020
Partner of
EuDML logo