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Boolean rings and Boolean
algebras, though historically and conceptually different, were shown by Stone to be
equationally interdefinable. Indeed, in a Boolean ring, addition can be defined
in terms of the ring multiplication and the successor operation (Boolean
complementation) x = 1 + x(= 1 − x). In this paper, it is shown that this type
of equational definability of addition also holds in a much wider class of
rings, namely periodic rings (ring satisfying xm = xn, m≠n) in which the
idempotent elements are “well behaved.” More generally, the following theorem is
proved:
Suppose R is a ring with unity 1, not necessarily commutative. Suppose
further that R satisfies the identity xn = xn+1f(x) where n is a fixed positive
integer and f(x) is a fixed polynomial with integer coefficients. If, further, the
idempotent elements of R commute with each other, then addition in R is
equationally definable in terms of multiplication in R and the successor operation
x = 1 + x.
Some new classes of rings to which this theorem applies are
exhibited.
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