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Algebra universalis

, Volume 73, Issue 2, pp 183–204 | Cite as

Idempotent generated algebras and Boolean powers of commutative rings

  • Guram Bezhanishvili
  • Vincenzo Marra
  • Patrick J. Morandi
  • Bruce Olberding
Article

Abstract

A Boolean power S of a commutative ring R has the structure of a commutative R-algebra, and with respect to this structure, each element of S can be written uniquely as an R-linear combination of orthogonal idempotents so that the sum of the idempotents is 1 and their coefficients are distinct. In order to formalize this decomposition property, we introduce the concept of a Specker R-algebra, and we prove that the Boolean powers of R are up to isomorphism precisely the Specker Ralgebras. We also show that these algebras are characterized in terms of a functorial construction having roots in the work of Bergman and Rota. When R is indecomposable, we prove that S is a Specker R-algebra iff S is a projective R-module, thus strengthening a theorem of Bergman, and when R is a domain, we show that S is a Specker R-algebra iff S is a torsion-free R-module.

For indecomposable R, we prove that the category of Specker R-algebras is equivalent to the category of Boolean algebras, and hence is dually equivalent to the category of Stone spaces. In addition, when R is a domain, we show that the category of Baer Specker R-algebras is equivalent to the category of complete Boolean algebras, and hence is dually equivalent to the category of extremally disconnected compact Hausdorff spaces.

For totally ordered R, we prove that there is a unique partial order on a Specker R-algebra S for which it is an f-algebra over R, and show that S is isomorphic to the R-algebra of piecewise constant continuous functions from a Stone space X to R equipped with the interval topology.

Key words and phrases

algebra over a commutative ring idempotent generated algebra Boolean power Stone space Baer ring f-ring Specker -group 

2010 Mathematics Subject Classification

Primary: 16G30 Secondary: 06E15 54H10 06F25 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Guram Bezhanishvili
    • 1
  • Vincenzo Marra
    • 2
  • Patrick J. Morandi
    • 1
  • Bruce Olberding
    • 1
  1. 1.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA
  2. 2.Dipartimento di Matematica “Federigo Enriques”Università degli Studi di MilanoMilanoItaly

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