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Generalized Radical Rings

  • W. Edwin Clark (a1)

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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, bR. 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.

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References

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1. Albert, A. A., Structure of algebras, Amer. Math. Soc. Coll. Publ., 24 (1939).
2. Clark, W. E. and Lewin, J., On minimal ideals in the circle composition semigroup of a ring to appear in Publ. Math. Debrecen).
3. Clifford, A. H., Semigroups containing minimal ideals, Amer. J. Math., 70 (1948), 521526.
4. Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vol. I, Math Surveys, No. 7 (Amer. Math. Soc, 1961).
5. Drazin, M. P., Rings with central idempotent or nilpotent elements, Proc. Edinburgh Math. Soc, 9 (1958), 157165.
6. Jacobson, N., Structure of rings, Amer. Math. Soc. Coll. Publ., 37 (1956).
7. Luh, J., A note on strongly regular rings, Proc. Japan Acad., 40, No. 2 (1964), 7475.
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