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.
Highlights
An element x of a ring R is regular if there exists y ∈ R such that x = xyx
A ring R is one-sided unit-regular if and only if for all finitely generated projective right R-modules A, B and C, A ⊕ B ∼= A ⊕ C implies that B ⊕ C or C ⊕ B
We prove that every matrix extension of an exchange locally comparable ideal is locally comparable
Summary
In [2], Chen proved that comparability of modules over one-sided unit-regular rings is Morita invariant, in terms of comparability. In [4], the author introduced and investigated a kind of quasi-stable exchange ideals These inspires us to explore local comparability depending only on the ring structure of an ideal and investigate certain matrix reduction over rings which might have no any comparability. Following Ara, an ideal I of a ring R is an exchange ideal provided that for every x ∈ I there exist an idempotent e ∈ I and elements r, s ∈ I such that e = xr = x + s − xs (cf [1]). M ⊕ N means that M is isomorphic to a direct summand of N
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