Non-simple purely infinite rings

TL;DR

This paper introduces the concept of purely infinite rings, extending the existing notion from simple rings, and explores their properties, examples, and stability under various algebraic operations.

Contribution

It defines purely infinite rings beyond the simple case and proves their invariance under Morita equivalence and other algebraic constructions.

Findings

Purely infinite rings include many important algebraic examples.

The property is preserved under matrix formation, corners, and extensions.

Tensor products with certain Leavitt path algebras are purely infinite.

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.