On uniquely clean rings. (English) Zbl 1251.16027
In this paper, the author studies uniquely clean (uc) and uniquely nil clean (unc) rings. He gets some equivalent conditions for a (uc) ring.
The author proves that a ring \(R\) is a (uc) ring if and only if any one of the following conditions holds.
The author has been very casual in giving references. At several places while quoting Theorem 2.1 he says it is Theorem 2.5. In proving Theorem 2.5, instead of saying “if” and “only if”, the author says “1 implies 2” and “2 implies 1”.
The author proves that a ring \(R\) is a (uc) ring if and only if any one of the following conditions holds.
- (i)
- \(R\) is an Abelian exchange ring whose factor rings modulo maximal ideals are 2-element fields.
- (ii)
- If \(J^*(R)\) denotes the intersection of all maximal ideals of \(R\) then \(R\) is an exchange ring such that \(R/J^*(R)\) is Boolean and idempotents can be lifted uniquely modulo \(J^*(R)\).
- (iii)
- \(R/J^*(R)\) is Boolean and \(R\) is strongly clean in the sense that every element of \(R\) is a sum of a central idempotent and a unit in \(R\).
The author has been very casual in giving references. At several places while quoting Theorem 2.1 he says it is Theorem 2.5. In proving Theorem 2.5, instead of saying “if” and “only if”, the author says “1 implies 2” and “2 implies 1”.
Reviewer: Veereshwar A. Hiremath (Dharwad)
MSC:
16U60 | Units, groups of units (associative rings and algebras) |
16E50 | von Neumann regular rings and generalizations (associative algebraic aspects) |
16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |
16N40 | Nil and nilpotent radicals, sets, ideals, associative rings |
Keywords:
exchange rings; Boolean rings; uniquely clean rings; uniquely nil clean rings; Abelian rings; central idempotents; nilpotent elementsReferences:
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