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Simple Cuntz–Pimsner rings

Authors
Journal
Journal of Algebra
0021-8693
Publisher
Elsevier
Publication Date
Volume
371
Identifiers
DOI: 10.1016/j.jalgebra.2012.08.005
Keywords
  • Cuntz–Pimsner Rings
  • Simplicity
  • Invariant Cycles
  • Condition (K)
  • Condition (L)
  • Cuntz–Krieger Uniqueness
  • Toeplitz Rings
  • Leavitt Path Algebras
  • Crossed Products
  • Fractional Skew Monoid Rings
Disciplines
  • Mathematics

Abstract

Abstract Necessary and sufficient conditions for when every non-zero ideal in a relative Cuntz–Pimsner ring contains a non-zero graded ideal, when a relative Cuntz–Pimsner ring is simple, and when every ideal in a relative Cuntz–Pimsner ring is graded, are given. A “Cuntz–Krieger uniqueness theorem” for relative Cuntz–Pimsner rings is also given and condition (L) and condition (K) for relative Cuntz–Pimsner rings are introduced. As applications of these results, a uniqueness result for the Toeplitz algebra of a directed graph and characterizations of when crossed products of a ring by a single automorphism and fractional skew monoid rings of a single corner isomorphism are simple, are obtained.

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