Vol. 74, No. 2, 1978

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Equational definability of addition in certain rings

Hal G. Moore and Adil Mohamed Yaqub

Vol. 74 (1978), No. 2, 407–417

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, mn) 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.

Mathematical Subject Classification
Primary: 16A38, 16A38
Secondary: 16A32
Received: 16 February 1977
Revised: 15 August 1977
Published: 1 February 1978
Hal G. Moore
Adil Mohamed Yaqub