Vol. 59, No. 2, 1975

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Infinite subrings of infinite rings and near-rings

Howard Edwin Bell

Vol. 59 (1975), No. 2, 345–358

Leffey has proved that every infinite associative ring contains an infinite commutative subring, and thereby suggested the problem of finding reasonably small classes of infinite rings with the property that () every infinite ring contains a subring belonging to . Clearly, there is no minimal class in the obvious sense, for in any class satisfying (*) a ring may be replaced by any proper infinite subring of itself. In §§1-3 we determine a class 0 satisfying (*) and consisting of familiar and easily-described rings; and § 4 we indicate how our results subsume and extend known finiteness results formulated in terms of subrings and zero divisors.

Section 5 identifies classes which satisfy (*) and are minimal in a certain loose sense, and § 6 extends the major result of the first three sections to distributive nearrings. The ring-theoretic results are proved in the setting of alternative rings.

Mathematical Subject Classification
Primary: 16A76
Received: 21 May 1975
Published: 1 August 1975
Howard Edwin Bell