|
Let (R,×,+) be a commutative
ring with unit 1, and let K = {ρ1,ρ2,⋯} be a transformation group in R. (R,×,+) is
called a ring-logic, mod K essentially if the “+” of R is equationally definable
in terms of the “K-logic” (R,×,ρ1,ρ2,⋯). The Boolean theory results by
choosing K to be the group generated by x∗ = 1 − x (order 2, x∗∗ = x). The
following result is proved: Let n = p1⋯pt be square-free, and let Rn be the
residue class ring, mod n. Let, ⌢, be any transitive 0 → 1 permutation
of Rpi(i = 1,⋯,t). Let, ⌢, be the induced permutation of Rn defined by
(x1,⋯,xt)⌢ = (x1⌢,⋯,xt⌢), xi ∈ Rpi(i = 1,⋯,t), and let K be the transformation
group in Rn generated by, ⌢. Then (Rn,×,+) is a ring-logic, mod K. An extension
of this theorem to the case where n is arbitrary is also considered. The present proofs
use the Fermat-Euler Theorem as well as a generalized form of the Chinese Residue
Theorem.
|
