Semidistributive and distributively decomposable rings
A module \(M_A\) is called semidistributive if it is a direct sum of distributive modules. A ring \(A\) is right distributively decomposable if its identity element is a sum of nonzero orthogonal idempotents \(e_1,\dots,e_n\) such that all \(e_iAe_i\) are right distributive rings. In the present note the right semidistributive rings, as well as distributively decomposable rings are investigated. If \(A\) is a Noetherian right distributively decomposable ring, then \(\bigcap_{i=1}^\infty J(A)^i=0\) and the right Krull dimension of \(A\) is less than or equal to 1. If \(A\) is a semiperfect semiprime right hereditary ring, then the following conditions are equivalent: (1) \(A\) is a right Noetherian ring; (2) \(A\) is a right serial ring; (3) \(A\) is a right semidistributive ring. Some results on the representation of the ring \(A\) as a finite direct product of right distributively decomposable rings are indicated.
semiperfect semiprime right hereditary rings
scientific article; zbMATH DE number 1407525