JACOBSON'S DENSITY THEOREM IN UNIVERSAL ALGEBRA

Abstract

This chapter elaborates the Jacobson's density theorem in universal algebra. Every Jacobson-semisimple ring is a subdirect product of primitive rings and that every primitive ring is, as a ring of operators, dense in his bicentralizer over some faithful left module. It is assumed that A is a universal algebra belonging to the variety V − V (Ω′, ∧′ ), where Ω′ is consisting not only of nullary and unary operations. It is supposed that π is an arbitrary, but fixed (μ + l)-ary operation of with Ω′ ≥ 1. Each left module M over A can be considered as an algebra (M, Ω ∪ A) , where the elements of A are considered as μ-ary operations. The homomorphisms of such a left module M will be called the A-homomorphisms of M; the corresponding congruences, the A-congruences of M, and the operations induced by the elements of A will be called A-left-multiplications of M.