\(K_0\)-theory of n-potents in rings and algebras

Abstract

Let \(n \geqslant 2\) be an integer. An n-potent is an element e of a ring R such that \(e^n = e\). We study n-potents in matrices over R and use them to construct an abelian group \(K_0^n(R)\). If A is a complex algebra, there is a group isomorphism \(K_0^n(A) \cong (K_0(A))^{n-1}\) for all \(n \geqslant 2\). However, for algebras over cyclotomic fields, this is not true, in general. We consider \(K_0^n\) as a covariant functor, and show that it is also functorial for a generalization of homomorphism called an n-homomorphism.

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Acknowledgements

The authors thank Dana Williams and Tom Shemanske for their helpful comments and suggestions.

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Correspondence to Jody Trout.

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Park, E., Trout, J. \(K_0\)-theory of n-potents in rings and algebras. European Journal of Mathematics 5, 395–410 (2019). https://doi.org/10.1007/s40879-018-0283-4

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Keywords

  • K-theory
  • n-Potents
  • Cyclotomic algebras

Mathematics Subject Classification

  • 18F30
  • 19A99
  • 19K99