MULTIPLICATIVE CLASSIFICATION OF ASSOCIATIVE RINGS

Let be a ring, and the left and right annihilators of the element , the two-sided ideal in called the additive controller, and let be an -isomorphism (i.e., multiplicative isomorphism) and its defect. An ideal in the ring is called an -ideal if for all -isomorphisms , is an ideal in and if and only if . It is shown that Very general sufficient conditions are given that a multiplicative isomorphism of subsemigroups of multiplicative semigroups of rings be extendible to the isomorphism of the subrings generated by them. Minimal prime ideals and the prime radical of a ring are -ideals. The strongly regular and regular rings that have unique addition are characterized.Bibliography: 29 titles.