Geodesic
Stable Subsets and the Existence of a Unit in (Semi)-Prime Rings
Matematičeskie zametki, Tome 70 (2001) no. 1, pp. 137-145

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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. 
@article{MZM_2001_70_1_a14,
     author = {A. V. Khokhlov},
     title = {Stable {Subsets} and the {Existence} of a {Unit} in {(Semi)-Prime} {Rings}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {137--145},
     publisher = {mathdoc},
     volume = {70},
     number = {1},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2001_70_1_a14/}
}
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A. V. Khokhlov. Stable Subsets and the Existence of a Unit in (Semi)-Prime Rings. Matematičeskie zametki, Tome 70 (2001) no. 1, pp. 137-145. http://geodesic.mathdoc.fr/item/MZM_2001_70_1_a14/