Simple Cuntz-Pimsner rings.
Relative Cuntz-Pimsner rings are algebraic analogues of relative Cuntz-Pimsner \(C^*\)-algebras. This kind of rings generalizes several algebraic structures among which we can mention Leavitt path algebras. As in the case of Leavitt path algebras, relative Cuntz-Pimsner rings come equipped with a \(\mathbb Z\)-grading. In a paper by the two first authors in 2011, a complete description of all graded ideals of an arbitraty Cuntz-Pimsner ring is given. In the present paper, necessary and sufficient conditions are given to ensure that a nonzero ideal in a relative Cuntz-Pimsner ring contains a nonzero-graded ideal. Then, simplicity of relative Cuntz-Pimsner rings is characterized. Also necessary and sufficient conditions are given to determine when every ideal in a relative Cuntz-Pimsner ring is graded. The authors also introduce the natural analogues of condition (L) and condition (K) in relative Cuntz-Pimsner rings, so as to get results extending the ones for the corresponding classes of Leavitt path algebras. Finally the paper deals with a uniqueness result for the Toeplitz algebra of a directed graph and with characterizations of simple crossed products of a ring by a single automorphism and simple fractional skew monoid rings of a single corner isomorphism.
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