Leffey has proved that every
infinite associative ring contains an infinite commutative subring, and thereby
suggested the problem of finding reasonably small classes ℐ of infinite rings with
the property that (∗) every infinite ring contains a subring belonging to ℐ.
Clearly, there is no minimal class ℐ 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 ℐ0 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.
