LOCAL COMPARABILITY OF EXCHANGE IDEALS

Year 2019, , 1 - 11, 08.01.2019

Handan Kose , Yosum Kurtulmaz Huanyin Chen

https://doi.org/10.24330/ieja.504095

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

References

• P. Ara, Extensions of exchange rings, J. Algebra, 197(2) (1997), 409-423.
• H. Chen, Elements in one-sided unit regular rings, Comm. Algebra, 25(8) (1997), 2517-2529.
• H. Chen, On generalized stable ideals, Comm. Algebra, 38(10) (2010), 3567-3579.
• H. Chen, On quasi-stable exchange ideals, J. Korean Math. Soc., 47(1) (2010), 1-15.
• H. Chen, Rings Related to Stable Range Conditions, Series in Algebra, 11, World Scientic Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.
• C. Huang, Refinement rings, exchange property and comparability, Bull. Korean Math. Soc., 48(3) (2011), 455-468.
• D. Khurana, T. Y. Lam and P. P. Nielsen, Exchange elements in rings, and the equation XA-BX = I, Trans. Amer. Math. Soc., 369(1) (2017), 495-516.
• F. Perera, Lifting units modulo exchange ideals and C*-algebras with real rank zero, J. Reine Angew. Math., 522 (2000), 51-62.