On certain finite rings and ring-logics
Adil Yaqub · Pacific Journal of Mathematics · 1962
Introduction.Boolean rings (B y x , +) and Boolean logics ( = Boolean algebras) (B, n, *) though historically and conceptionally different, are equationally interdefinable in a familiar way [6].With this equational interdefinability as motivation, Foster introduced and studied the theory of ring-logics.In this theory, a ring (or an algebra) R is studied modulo K, where K is an arbitrary transformation group in R. The Boolean theory results from the special choice, for K, of the "Boolean group", generated by x* = 1x (order 2, #** = x).More generally, in a commutative ring (R, x, +) with identity 1, the natural group N, generated by αΓ= 1 + x (with αf= x -1 as inverse) proved to be of particular interest.Thus, specialized to N, a commutative ring with identity (R, x, +) is called a ring-logic, mod N if (1) the + of the ring is equationally definable in terms of its TV-logic (R, x, ~, ^), and (2) the + of the ring is fixed by its ΛΓ-logic.Several classes of ring-logics (modulo suitably chosen groups) are known [1; 2; 7], and the object of this manuscript is to extend further the class of ring-logics.Indeed, we shall prove the following: THEOREM 1.Let R be any finite commutative ring with zero radical.Then, R is a ring-logic, mod N.