Vol. 15, No. 4, 1965

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Ring-logics and residue class rings

Adil Mohamed Yaqub

Vol. 15 (1965), No. 4, 1465–1469

Let (R,×,+) be a commutative ring with unit 1, and let K = {ρ12,} 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,×12,). 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 = p1pt 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.

Mathematical Subject Classification
Primary: 16.99
Received: 6 July 1964
Published: 1 December 1965
Adil Mohamed Yaqub