Non-simple purely infinite rings

Abstract

In this paper we introduce the concept of purely infinite rings, which in the simple case agrees with the already existing notion of pure infiniteness. We establish various permanence properties of this notion, with respect to passage to matrix rings, corners, and behaviour under extensions, so being purely infinite is preserved under Morita equivalence. We show that a wealth of examples falls into this class, including important analogues of constructions commonly found in operator algebras. In particular, for any (s-) unital K-algebra having enough nonzero idempotents (for example, for a von Neumann regular algebra) its tensor product over K with many nonsimple Leavitt path algebras is purely infinite. G. A. P. and M. S. M. were supported in part by the Spanish MEC and Fondos FEDER through project MTM2007-60333, and jointly by the Junta de Andalucía and Fondos FEDER through projects FQM-336 and FQM-2467; F. P. was supported in part by the DGI MEC-FEDER through Project MTM2005-00934 and the