Commutative rings for which every continuous module is quasi-injective

Abstract

1. Introduction. The concept of continuous modules owes its origin to Von Neumann's work on continuous geometries [16], and Utumi's work on continuous rings ([12], [13], [14]). L. Jeremy in [3] defined continuous modules, and others ([2], [5], [6], [7], [10]) later provided many interesting properties of such modules. A module is called continuous if (i) every submodule is essential in a summand and (ii) every submodule, isomorphic to a summand, is itself a summand. It can be easily seen, that while every quasi-injective module is continuous ([3], [5], [10]), the converse is not true. It is, therefore, natural to ask: For what classes of rings, is every continuous module quasi-injective? We provide here some classes of commutative rings, for which this is true. Our study is motivated by a general interest in identifying rings for which continuity is a non-trivial concept, and specifically, because only for such rings the existence of continuous hulls [7] poses a problem. We show that large classes of commutative rings, such as noetherian ones, and semiprimary ones with Jacobson radical square zero, have the property that every continuous module is quasi-injective. We give a necessary and sufficient condition, over an arbitrary commutative ring, for every uniform continuous module to be quasi-injective. We also provide necessary and sufficient conditions for some special classes of commutative rings, such as perfect rings, finitary rings, and chain rings, to have the property that every continuous module is quasi-injective. All our rings are commutative, have identity elements, and all modules are unitary rightmodules. Homomorphisms and endomorphisms act on the left. E(M)= EM denotes the injective hull of a module M. A c'M means that A is an essential submodule of M, a "summand" means a "direct summand", and e ~ stands for the annihilator in R of an element e of an R-module. A module A is said to be B-injective if every homomorphism from a submodule of B to A can be extended to B. We recall that a module M is called quasi-continuous (or 7r-injective) if M is closed under all idempotents of EndoRE(M) ([2], [3]). This holds if and only if every decomposition E(M)-- ~ @ E, induces a decomposition M= ~ @ (M hE,) tel tel ([2], Theorem 1.1). Every continuous module is quasi-continuous, and every quasi-continuous module satisfies the condition (i) of the definition of a continuous module. Let Mbe a module, with an injective hull E, then a continuous overmodule C of Mis called a continuous hull of M if Mc C ~ E, and if C c X holds for every continuous module M= X c E. We mention that this definition is independent of the choice of E [7].