Fundamentals

Abstract

An element r of a ring R is called a

• left (right) zero divisor if there is a nonzero s ∈ R : rs = 0(sr = 0);

• zero divisor if it is left and right divisor;

• left (right) cancellable if for every a, b ∈ R : ra = rb(ar = br) ⇒ a = b;

• idempotent if r ² = r; two idempotents e, e’ are orthogonal if ee’ = e’e = 0; in a ring R we denote by Id(R) the set of all the idempotent elements. A ring is called Boole if all its elements are idempotent.