Groupes et Anneaux Réticulés

Bigard, Alain; Keimel, Klaus; Wolfenstein, Samuel

doi:10.1007/bfb0067004

Cited by 455 publications

(336 citation statements)

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“…As we see above the idea of a clean ring is completely independent of this extra structure. We shall lay the groundwork here but for additional information we suggest the reader check Darnel (1995) or the classic reference Bigard et al (1977).…”

Section: One Of the Main Results Is The Followingmentioning

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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Section: One Of the Main Results Is The Followingmentioning

confidence: 99%

Clean Semiprimef-Rings with Bounded Inversion

McGovern

2003

Communications in Algebra

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“…Now, let A be an /-algebra with respect to a multiplication *. In the next theorem, we will establish a relationship between the product / * g of two elements f,g G A and their product fg in the /-algebra A u , the universal completion of A. P r o o f. It is well-known that the multiplication * in A extends uniquely to an /-algebra multiplication in the Dedekind completion A s of A in such a manner that A becomes a subalgebra of A s (for details see [4], [11]). Furthermore, since A is an order dense vector sublattice of A u and A u is Dedekind complete, A 6 is also an order dense vector sublattice of A u .…”

Section: Preliminariesmentioning

confidence: 99%

On the Structure of Almost F-Algebras

Boulabiar¹,

Chil²

2001

Demonstratio Mathematica

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“…General references for /-groups (i.e., lattice-ordered groups) are [5,1,13] (though this last is about vector lattices) and the recent compendium [9].…”

Section: Preliminariesmentioning

confidence: 99%

Epimorphic Adjunction of a Weak Order Unit to an Archimedean Lattice-Ordered Group

Ball¹,

Hager²,

Kizanis³

1992

Proceedings of the American Mathematical Society

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Abstract.It is shown that an archimedean /-group G can be embedded into another, H, which has a weak unit, by an embedding that is epimorphic in archimedean /-groups if and only if there is countable A Ç G with A1-= (0). Then the extension H can always be chosen conditionally and laterally acomplete and the embedding essential, but can never be generated by G together with finitely many extra elements unless G already had a weak unit.

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Groupes et Anneaux Réticulés

Cited by 455 publications

References 0 publications

Clean Semiprimef-Rings with Bounded Inversion

Clean Semiprimef-Rings with Bounded Inversion

On the Structure of Almost F-Algebras

Epimorphic Adjunction of a Weak Order Unit to an Archimedean Lattice-Ordered Group

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