  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.
