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

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 $r\in Rr$ for any $r\in R$ is a ring with unit. In this connection, examples of prime (and even simple) rings are presented such that $r\in Rr\cap rR$ for any $r\in R$ 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 $r\in Rr$ for any element $r$ of the prime radical $P(R)$ and the ring $R$ $P(R)$ has a left unit.