Vol. 19, No. 1, 1966

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ISSN: 0030-8730
Some classes of ring-logics

Adil Mohamed Yaqub

Vol. 19 (1966), No. 1, 189–195

Let (R,×,+) be a commutative ring with identity, and let K = {ρ12,} be a transformation group in R. The K-logic of the ring (R,×,+) is the (operationally closed) system (R,×12,) whose operations are the ring product “×” together with the unary operations ρ12, of K. The ring (R,×,+) is essentially a ring-logic, mod K, if the “+” of the ring is equationally definable in terms of its K-logic (R,×12,). Our present object, is to show that any finite direct product of (not necessarilly finite) direct powers of finite commutative local rings of distinct orders is a ring-logic modulo certain suitably chosen (but nevertheless still rather general) groups. This theorem subsumes and generalizes Foster’s results for Boolean rings, p-rings, and pk-rings, as well as the author’s results for residue class rings and finite commutative rings with zero radical. Several new classes of ring-logics (modulo certain groups of quite general nature) are also explicitly exhibited. Throughout the entire paper, all rings under consideration are assumed to be commutative and with identity.

Mathematical Subject Classification
Primary: 16.99
Secondary: 02.99
Received: 15 June 1965
Published: 1 October 1966
Adil Mohamed Yaqub