LOCAL COMPARABILITY OF EXCHANGE IDEALS

Abstract

An exchange ideal $I$ of a ring $R$ is locally comparable if

for every regular $x\in I$ there exists a right or left invertible $u\in 1+I$ such

that $x=xux$. We prove that every matrix extension

of an exchange locally comparable ideal is locally comparable. We thereby

prove that every square regular matrix over such ideal admits a

diagonal reduction.

Keywords

• Locally comparable ideal,matrix extension,diagonal reduction,exchange ideal

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