Strong separativity over exchange rings

Huanyin Chen

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 2, page 417-428
  • ISSN: 0011-4642

Abstract

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An exchange ring R is strongly separative provided that for all finitely generated projective right R -modules A and B , A ⊕ A ≅ A ⊕ B ⇒ A ≅ B . We prove that an exchange ring R is strongly separative if and only if for any corner S of R , a S + b S = S implies that there exist u , v ∈ S such that a u = b v and S u + S v = S if and only if for any corner S of R , a S + b S = S implies that there exists a right invertible matrix a b * ∈ M 2 ( S ) . The dual assertions are also proved.

How to cite

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Chen, Huanyin. "Strong separativity over exchange rings." Czechoslovak Mathematical Journal 58.2 (2008): 417-428. <http://eudml.org/doc/31218>.

@article{Chen2008,
abstract = {An exchange ring $R$ is strongly separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A \oplus B\Rightarrow A\cong B$. We prove that an exchange ring $R$ is strongly separative if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exist $u,v\in S$ such that $au=bv$ and $Su+Sv=S$ if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exists a right invertible matrix $\begin\{pmatrix\} a&b\\ *&* \end\{pmatrix\} \in M_2(S)$. The dual assertions are also proved.},
author = {Chen, Huanyin},
journal = {Czechoslovak Mathematical Journal},
keywords = {strong separativity; exchange ring; regular ring; strong separativity; exchange rings; regular rings},
language = {eng},
number = {2},
pages = {417-428},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Strong separativity over exchange rings},
url = {http://eudml.org/doc/31218},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Chen, Huanyin
TI - Strong separativity over exchange rings
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 2
SP - 417
EP - 428
AB - An exchange ring $R$ is strongly separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A \oplus B\Rightarrow A\cong B$. We prove that an exchange ring $R$ is strongly separative if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exist $u,v\in S$ such that $au=bv$ and $Su+Sv=S$ if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exists a right invertible matrix $\begin{pmatrix} a&b\\ *&* \end{pmatrix} \in M_2(S)$. The dual assertions are also proved.
LA - eng
KW - strong separativity; exchange ring; regular ring; strong separativity; exchange rings; regular rings
UR - http://eudml.org/doc/31218
ER -

References

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