PIERCE STALKS OF EXCHANGE RINGS
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We prove, in this article, that a ring R is a stable exchange ring if and only if so are all its Pierce stalks. If every Pierce stalks of R is artinian, then 1R = u + v with u, v ∈ U(R) if and only if for any a ∈ R, there exist u, v ∈ U(R) such that a = u + v. Furthermore, there exists u ∈ U(R) such that 1R ± u ∈ U(R) if and only if for any a ∈ R,there exists u ∈ U(R) such that a ± u ∈ U(R). We will give analogues to normal exchange rings. The root properties of such exchange rings are also obtained.
Abstract
We prove, in this article, that a ring R is a stable exchange ring if and only if so are all its Pierce stalks. If every Pierce stalks of R is artinian, then 1R = u + v with u, v ∈ U(R) if and only if for any a ∈ R, there exist u, v ∈ U(R) such that a = u + v. Furthermore, there exists u ∈ U(R) such that 1R ± u ∈ U(R) if and only if for any a ∈ R,there exists u ∈ U(R) such that a ± u ∈ U(R). We will give analogues to normal exchange rings. The root properties of such exchange rings are also obtained.
참고문헌 (12)
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