# Idempotent generated algebras and Boolean powers of commutative rings

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## 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

*ℓ*-group

## 2010 Mathematics Subject Classification

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

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