Malcolmson semigroups

Fun, Hung, Tsz; Li, Hanfeng

doi:10.1016/j.jalgebra.2023.01.031

Journal of Algebra

2023

DOI: 10.1016/j.jalgebra.2023.01.031

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Malcolmson semigroups

Hung, Tsz Fun

Hanfeng Li

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2024

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The Cuntz semigroup of a ring

Antoine,

Ara,

Bosa

et al. 2024

Sel. Math. New Ser.

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For any ring R, we introduce an invariant in the form of a partially ordered abelian semigroup $${\mathrm S}(R)$$ S ( R ) built from an equivalence relation on the class of countably generated projective modules. We call $${\mathrm S}(R)$$ S ( R ) the Cuntz semigroup of the ring R. This construction is akin to the manufacture of the Cuntz semigroup of a C*-algebra using countably generated Hilbert modules. To circumvent the lack of a topology in a general ring R, we deepen our understanding of countably projective modules over R, thus uncovering new features in their direct limit decompositions, which in turn yields two equivalent descriptions of $${\mathrm S}(R)$$ S ( R ) . The Cuntz semigroup of R is part of a new invariant $$\textrm{SCu}(R)$$ SCu ( R ) which includes an ambient semigroup in the category of abstract Cuntz semigroups that provides additional information. We provide computations for both $${\mathrm S}(R)$$ S ( R ) and $$\textrm{SCu}(R)$$ SCu ( R ) in a number of interesting situations, such as unit-regular rings, semilocal rings, and in the context of nearly simple domains. We also relate our construcion to the Cuntz semigroup of a C*-algebra.

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The Cuntz semigroup of a ring

Antoine,

Ara,

Bosa

et al. 2024

Sel. Math. New Ser.

View full text Add to dashboard Cite

show abstract

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Malcolmson semigroups

Cited by 1 publication

References 32 publications

The Cuntz semigroup of a ring

The Cuntz semigroup of a ring

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