Quasi-Clean Rings

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Thesis Type: Postgraduate

Institution Of The Thesis: Erzincan Binali Yildirim University, Fen Bilimleri Enstitüsü, MATEMATİK ANABİLİM DALI, Turkey

Approval Date: 2024

Thesis Language: Turkish

Student: HAZEL DUMLU POLAT

Supervisor: Tufan Özdin

Open Archive Collection: AVESIS Open Access Collection

Abstract:

Exchange rings have been characterized by the property that, for all a∈R, there exists an idempotent e∈aR such that 1-e∈(1-a)R. In this thesis, we introduce the notion of quasi-exchange rings via quasi-idempotent elements. We prove that a ring R is quasi-exchange if and only if, given a right ideal I of a ring R and x∈R with kx-x^2∈I where k is a central unit of R, there exist a quasi-idempotent q^2=kq such that q-x∈I. We also show that R is a quasi-exchange ring if and only if R⁄(J(R)) a quasi-exchange ring and R satisfies the previons equivalent condition for the Jacobson radical J(R). Some relations between quasi-exchange rings and (strongly) quasi-clean rings are also obtained.