Stable Subsets and the Existence of a Unit in (Semi)-Prime Rings

Abstract

Criteria for the existence of a unit in a semiprime, prime, or simple ring and criteria for an idempotent of an arbitrary ring or of a semiprime ring to be central are obtained. In particular, it is shown that a strictly prime ring R in which r ∈ Rr for any r ∈ R is a ring with unit. In this connection, examples of prime (and even simple) rings are presented such that r ∈ Rr ∩ rR for any r ∈ R but there is no unit. The problem of whether a given ring R has a left unit was reduced earlier by the author to the semiprime case, namely, R has a left unit if and only if r ∈ Rr for any element r of the prime radical P(R) and the ring RP(R) has a left unit.