Distributive rings and modules

Abstract

DISTRIBUTIVE RINGS AND MODULES A. A. Tuganbaev All rings are assumed to be associative with a nonzero unity, and all modules are uni- tal, and unless it is stated otherwise, right modules. We denote by L(M) the lattice of all submodules of the module M. A module M is called a distributive module (chain module) if L(M) is a distributive lattice (chain). Words like "distributive ring" mean that the corresponding conditions hold on the right. THEOREM i. In a distributive ring in which all nilpotent elements are central, the intersection of any two finitely generated right ideals is finitely generated. Theorem 1 follows from Proposition 2 and generalizes the analogous result for commu- tative distributive rings [i]. If ~ is a monomorphism of a ring A into itself, we denote by A [Ix, q,]] the ring of left (right) skew formal series in x, in which coefficients are written on the left (right) of x and multiplication is defined by the rule a a = ~p(~)x (ax -:xq (a)). THEOREM 2. Suppose ~ is a monomorphism of a ring A into itself and R------A [Ix, ~]]. Then the following conditions are equivalent: (a) R is a distributive ring; (b) R is a Bezout ring, and all right annihilators in A are ideals; (c) R is a Bezout ring, and all maximal right ideals of A are ideals; (d) A is a strongly regular, right and left countably injective ring, ~p is an auto- morphism, and ~ (t) = t for each idempotent t in A. The case where ,~ is an automorphism was considered earlier in [2]. By a Bezout module we mean a module with cyclic finitely generated submodules. A ring is called re- duced if it contains no nonzero nilpotent elements. A module M A is called countably injec- tive if each homomorphism BA--~3f, where B is any eountably generated right ideal of A can be extended to a homomorphism AA --> ~f. We denote by rA (B) (l~ (B)) the right (left) annihi- lator in A of the subset B. We denote by max (A) the set of maximal right ideals, and by J(A) the Jacobson radical, of the ring A. A subset T of A is called a set of right denom- inators if there exist a ring A T and a canonical homomorphism /T----/: A --, AT, such that all elements of f(T) are invertible in AT. Ar = {/(a)/(t) -I ]a ~A, ! ~ T}. Ker (/) = {a ~A Ir (a) (~ T ~-~}. Under these conditions, for any module N A there is defined an A-module homomor- phism gT--1@~: N-~A~-N ~A AT. The ring A is called localizable if A ~Mis a set of right denominators for all M in max (A). In this case we write AM, /M. ~I instead of AT, /T. gT. A module N A over a localizable ring A is locally chained if N M is a chain module over A M for all M~max (A). PP. Moscow Energetics Institute. Translated from Matematicheskie Zametki, Vol. 47, No. 2, 115-123, February, 1990. Original article submitted June 26, 1987. 0001-4346/90/4712-0199512.50 9 1990 Plenum Publishing Corporation 199