Exchange rings with stable range one

Huanyin Chen

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 2, page 579-590
  • ISSN: 0011-4642

Abstract

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We characterize exchange rings having stable range one. An exchange ring R has stable range one if and only if for any regular a ∈ R , there exist an e ∈ E ( R ) and a u ∈ U ( R ) such that a = e + u and a R ∩ e R = 0 if and only if for any regular a ∈ R , there exist e ∈ r . a n n ( a + ) and u ∈ U ( R ) such that a = e + u if and only if for any a , b ∈ R , R / a R ≅ R / b R ⟹ a R ≅ b R .

How to cite

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Chen, Huanyin. "Exchange rings with stable range one." Czechoslovak Mathematical Journal 57.2 (2007): 579-590. <http://eudml.org/doc/31148>.

@article{Chen2007,
abstract = {We characterize exchange rings having stable range one. An exchange ring $R$ has stable range one if and only if for any regular $a\in R$, there exist an $e\in E(R)$ and a $u\in U(R)$ such that $a=e+u$ and $aR\cap eR=0$ if and only if for any regular $a\in R$, there exist $e\in r.ann(a^+)$ and $u\in U(R)$ such that $a=e+u$ if and only if for any $a,b\in R$, $R/aR\cong R/bR\Longrightarrow aR\cong bR$.},
author = {Chen, Huanyin},
journal = {Czechoslovak Mathematical Journal},
keywords = {exchange ring; stable range one; idempotent; unit; exchange rings; stable range one; idempotents; units},
language = {eng},
number = {2},
pages = {579-590},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Exchange rings with stable range one},
url = {http://eudml.org/doc/31148},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Chen, Huanyin
TI - Exchange rings with stable range one
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 2
SP - 579
EP - 590
AB - We characterize exchange rings having stable range one. An exchange ring $R$ has stable range one if and only if for any regular $a\in R$, there exist an $e\in E(R)$ and a $u\in U(R)$ such that $a=e+u$ and $aR\cap eR=0$ if and only if for any regular $a\in R$, there exist $e\in r.ann(a^+)$ and $u\in U(R)$ such that $a=e+u$ if and only if for any $a,b\in R$, $R/aR\cong R/bR\Longrightarrow aR\cong bR$.
LA - eng
KW - exchange ring; stable range one; idempotent; unit; exchange rings; stable range one; idempotents; units
UR - http://eudml.org/doc/31148
ER -

References

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