Left APP-property of formal power series rings

Zhongkui Liu; Xiao Yan Yang

Archivum Mathematicum (2008)

  • Volume: 044, Issue: 3, page 185-189
  • ISSN: 0044-8753

Abstract

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A ring R is called a left APP-ring if the left annihilator l R ( R a ) is right s -unital as an ideal of R for any element a ∈ R . We consider left APP-property of the skew formal power series ring R [ [ x ; α ] ] where α 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 ; α ] ] is left APP if and only if for any sequence ( b 0 , b 1 , ⋯ ) of elements of R the ideal l R ( ∑ j = 0 ∞ ∑ k = 0 ∞ R α k ( b j ) ) 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.

How to cite

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Liu, Zhongkui, and Yang, Xiao Yan. "Left APP-property of formal power series rings." Archivum Mathematicum 044.3 (2008): 185-189. <http://eudml.org/doc/250493>.

@article{Liu2008,
abstract = {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.},
author = {Liu, Zhongkui, Yang, Xiao Yan},
journal = {Archivum Mathematicum},
keywords = {left APP-ring; skew power series ring; left principally quasi-Baer ring; left APP-rings; skew power series rings; left principally quasi-Baer rings; descending chain condition on right annihilators},
language = {eng},
number = {3},
pages = {185-189},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Left APP-property of formal power series rings},
url = {http://eudml.org/doc/250493},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Liu, Zhongkui
AU - Yang, Xiao Yan
TI - Left APP-property of formal power series rings
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 3
SP - 185
EP - 189
AB - 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.
LA - eng
KW - left APP-ring; skew power series ring; left principally quasi-Baer ring; left APP-rings; skew power series rings; left principally quasi-Baer rings; descending chain condition on right annihilators
UR - http://eudml.org/doc/250493
ER -

References

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