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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Lifting idempotents and exchange rings

Author: W. K. Nicholson
Journal: Trans. Amer. Math. Soc. 229 (1977), 269-278
MSC: Primary 16A32
MathSciNet review: 0439876
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Abstract: Idempotents can be lifted modulo a one-sided ideal L of a ring R if, given $ x \in R$ with $ x - {x^2} \in L$, there exists an idempotent $ e \in R$ such that $ e - x \in L$. Rings in which idempotents can be lifted modulo every left (equivalently right) ideal are studied and are shown to coincide with the exchange rings of Warfield. Some results of Warfield are deduced and it is shown that a projective module P has the finite exchange property if and only if, whenever $ P = N + M$ where N and M are submodules, there is a decomposition $ P = A \oplus B$ with $ A \subseteq N$ and $ B \subseteq M$.

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Keywords: Exchange property, exchange rings, lifting idempotents, von Neumann regular rings, projective modules
Article copyright: © Copyright 1977 American Mathematical Society