Divisible modules

Abstract

DIVISIBLE MODULES eben matlisIntroduction.Let R be an integral domain with quotient field Q, and let A be a module over P. A is said to be a divisible P-module, if rA =A for every r^OGi?.An element xG^4 is said to be a torsion element oí A, if there exists r^OGi?such that rx = 0.The set of torsion elements of A is a submodule of A called the torsion submodule of A, and we will consistently denote it by AT.We will let E(A) denote the infective envelope of A (see [3]) ; and hd« A will denote the homological dimension of ^4 as an P-module.We will study conditions, some necessary, some sufficient, for the torsion submodule of a divisible module to be a direct summand.These will be related to the condition that hdflÇ = l, where Q is the quotient field of P. We will apply these conditions to show that, if R is a Noetherian integral domain in which prime ideals different from zero are maximal, and if D is a divisible module over P, then D is a homomorphic image of an injective P-module and Dt is a direct summand of D. The same conclusions hold, if we merely assume for an arbitrary integral domain that its quotient field is countably generated as a module over the ring.