Separative exchange rings in which 2 is invertible

2014/08/07 by Huanyin Chen, Chen, Huanyin
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1408.1687

openalex publication_date 2014/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An exchange ring R is separative provided that for all finitely generated projective right R-modules A and B, A⊕ A≅ A⊕ B≅ B⊕ B\Longrightarrow A≅ B. Let R be a separative exchange ring in which 2 is invertible, and let a-a³∈ R be regular. We prove, in this note, that a∈ R is unit-regular if R(1-a²)R=Rr(a)=ℓ(a). An element a in a ring R is special clean if there exists an idempotent e∈ R such that a-e∈ R is a unit and aR\bigcap eR=0. Furthermore, we prove that a∈ R is special clean if aR/ar(a²), R/(aR+r(a)) are projective, and R(a-a³)R=Rar(a²)=ℓ (a²)aR. These also extend the corresponding results in separative regular rings.

Related

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• Unit-Regularity of Regular Nilpotent Elements
• K₁ of separative exchange rings and C*-algebras with real rank zero
• On some constructions of nil-clean, clean, and exchange rings
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