Distributively decomposable rings

Abstract

To prove the theorem we need some propositions and definitions. A proper ideal F of a ring A is said to be completely prime if A/F is a domain. An element α of Λ is said to be strictly nilpotent in A if all the terms of any sequence {an} with αχ = a and αη+ι 6 anAan are equal to zero, beginning with some index; P(A) coincides with the set of all strongly nilpotent elements of A. A semidistributive module is defined to be a direct sum of distributive modules. A ring A is said to be distributively decomposable if its identity is a sum of orthogonal idempotents e i , . . . ,en such that all the eiAe* are distributive rings. A module Μ A is said to be non-singular if the annihilators of all its non-zero elements are not essential right ideals of A. A ring without non-zero nilpotent elements is called a reduced ring.