Commutative Rings Whose Elements Are a Sum of a Unit and Idempotent

ABSTRACT. A ring with identity is said to be clean if every element can be written as a sum of a unit and an idempotent. The study of clean rings has been at the forefront of ring theory over the past decade. The theory of partially-ordered groups has a nice and long history and since there are several ways of relating a ring to a (unital) partially-ordered group it became apparent that there ought to be a notion of a clean partially-ordered group. In this article we define a clean unital lattice-ordered group; we state and prove a theorem which characterizes clean unital -groups. We mention the relationship of clean unital -groups to algebraic K-theory. In the last section of the article we generalize the notion of clean to the non-unital context and investigate this concept within the framework of W-objects, that is, archimedean -groups with distinguished weak order unit.

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“…Let X = N and let K be a compactification of X for which K X is homeomorphic to [0,1]. That such a space exists follows from [11:…”

Section: Clean Unital -Groups Clean Unital -Groupsmentioning

confidence: 99%

Clean unital ℓ-groups

Hager

Kimber

McGovern

2013

Mathematica Slovaca

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“…(1). It is routine to verify that RC 2k ∼ = (RC k )C 2 , so it suffices to show that if RC 2 is uniquely clean.…”

Section: Proving (2)mentioning

confidence: 99%

Rings in Which Elements Are Uniquely the Sum of an Idempotent and a Unit

Nicholson¹,

Zhou²

2004

Glasgow Math. J.

115

102

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Abstract. An associative ring with unity is called clean if every element is the sum of an idempotent and a unit; if this representation is unique for every element, we call the ring uniquely clean. These rings represent a natural generalization of the Boolean rings in that a ring is uniquely clean if and only if it is Boolean modulo the Jacobson radical and idempotents lift uniquely modulo the radical. We also show that every image of a uniquely clean ring is uniquely clean, and construct several noncommutative examples.

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“…The ring itself is called clean if every element is clean. Recently, Anderson and Camillo (Anderson, D. D., Camillo, V. (2002). Commutative rings whose elements are a sum of a unit and an idempotent.…”

mentioning

confidence: 99%

Clean Semiprimef-Rings with Bounded Inversion

McGovern

2003

Communications in Algebra

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An element in a ring is called clean if it may be written as a sum of a unit and idempotent. The ring itself is called clean if every element is clean. Recently, Anderson and Camillo (Anderson, D. D., Camillo, V. (2002). Commutative rings whose elements are a sum of a unit and an idempotent. Comm. Algebra 30(7):3327-3336) has shown that for commutative rings every von-Neumann regular ring as well as zero-dimensional rings are clean. Moreover, every clean ring is a pm-ring, that is every prime ideal is contained in a unique maximal ideal. In the same article, the authors give an example of a commutative ring which is a pm-ring yet not clean, e.g., C(R). It is this example which interests us. Our discussion shall take place in a more general setting. We assume that all rings are commutative with 1.

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Commutative Rings Whose Elements Are a Sum of a Unit and Idempotent

Cited by 109 publications

References 4 publications

Clean unital ℓ-groups

Clean unital ℓ-groups

Rings in Which Elements Are Uniquely the Sum of an Idempotent and a Unit

Clean Semiprimef-Rings with Bounded Inversion

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