Geodesic
Generalized Radical Rings
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 88-94

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Let R be a ring. We denote by o the so-called circle composition on R, denned by a o b = a + b — ab for a, b ∊ R. It is well known that this composition is associative and that R is a radical ring in the sense of Jacobson (see 6) if and only if the semigroup (R, o) is a group. We shall say that R is a generalized radical ring if (R, o) is a union of groups. Such rings might equally appropriately be called generalized strongly regular rings, since every strongly regular ring satisfies this property (see Theorem A below). This definition was in fact partially motivated by the observation of Jiang Luh (7) that a ring is strongly regular if and only if its multiplicative semigroup is a union of groups. 
Clark, W. Edwin. Generalized Radical Rings. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 88-94. doi: 10.4153/CJM-1968-011-2
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