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Let (R,×,+) be a commutative
ring with identity, and let K = {ρ1,ρ2,⋯} be a transformation group in R. The
K-logic of the ring (R,×,+) is the (operationally closed) system (R,×,ρ1,ρ2,⋯)
whose operations are the ring product “×” together with the unary operations
ρ1,ρ2,⋯ 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,×,ρ1,ρ2,⋯). 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.
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