Primitively pure submodules and primitively divisible modules

Abstract

A. A. Tuganbaev UDC 512.55All rings are assumed to be associative and to have a nonzero identity element. Let A be a ring,and let M be a rightA-module. A proper ideal P ofthe ringA is said to be right primitive if P is theannihilator ofa simple rightA-module (i.e., ifthere exists a maximal right idealB ofthe ringA suchthat P is the largest ideal of A contained in B). Consequently, every maximal ideal is a right and leftprimitive ideal. A submodule N ofthe moduleM is said to be primitively pure (resp. maximally pure)ifNP = NMP for every right primitive (resp. maximal) idealP ofthe ringA. A module M is said tobe primitively divisible (resp. maximally divisible)ifM = MP for every right primitive (resp. maximal)ideal P ofthe ringA.AringA is called a right max ring ifevery nonzero rightA-module has a maximalsubmodule. Maximally pure submodules and maximally divisible modules are studied in [3,4,6,7]. In [5],right max rings with polynomial identity are described. In [10], right quasi-invariant right max rings aredescribed.The main results ofthis paper are Theorems 1 and 2.Theorem1. Let A be either a P.I. ring or a right quasi-invariant ring. Then the following conditionsare equivalent.(1) A is a regular ring.(2) All submodules of any right A-module are primitively pure.(3) All submodules of any right A-module are maximally pure.(4) All principal right ideals of the ring A are primitively pure.(5) All principal right ideals of the ring A are maximally pure.Remark1. Let A be a simple ring. It is obvious that all submodules ofanyA-module M are maximallypure and primitively pure in M. Since the ring A is not necessarily regular, Theorem 1 does not hold forarbitrary rings.Theorem2. For a P.I. ring A,the following conditions are equivalent.(1) All prime ideals of the ring A are maximal ideals,andJ(A) is a right or left t-nilpotent ideal.(2) There do not exist nonzero primitively divisible right A-modules.(3) There do not exist nonzero primitively divisible left A-modules.(4) There do not exist nonzero maximally divisible right A-modules.(5) There do not exist nonzero maximally divisible left A-modules.(6) A is a right max ring.(7) A is a left max ring.The paper also contains some other assertions on primitively (resp. maximally) pure submodulesand primitively (resp. maximally) divisible modules.The proofs of Theorems 1 and 2 are decomposed into a series of lemmas. We present the necessarynotation and definitions. For a module M,theJacobson radical of M is denoted by J(M). For a moduleM and its subset X,wedenotebyr(X) the annihilator of X. For a module M, a submodule N ofM is said to be essential if NX = 0 for every nonzero submoduleX ofthe moduleM.AringA issaid to be regular if a ∈ aAa for each element a ∈ A.AringA is said to be right quasi-invariant if