COMULTIPLICATION MODULES OVER COMMUTATIVE RINGS

Let R be a commutative ring with identity. A unital H-module M is a comultiplication module provided for each submodule N of M there exists an ideal A of R such that N is the set of elements M in M such that Am = 0. It is proved that if M is a finitely generated comultiplication H-module with annihilator B in R then the ring RIB is semilocal and in certain cases M is quotient finite dimensional. Moreover, certain comultiplication modules satisfy the AB5*-condition. If an H-module X = circle plus U-i is an element of I(i) is a direct sum of simple submodules U-i (i is an element of I) and if P-i is the annihilator of U-i in R for each i in I then X is a comultiplication module if and only if boolean AND P-j not equal i(j) not subset of P-i for all i is an element of I. A Noetherian comultiplication module is Artinian and a finitely generated Artinian module M is a comultiplication module if and only if the socle of M is a (finite) direct sum of pairwise non-isomorphic simple submodules. In case R is a Dedekind domain, an H-module M is a comultiplication module if and only if M is cocyclic or M congruent to (R/P-1(k(1))) circle plus ... circle plus (R/P-n(k(n))) for some positive integers n, k(i) (1(1) <= i <= n) and distinct maximal ideals P-i (1 <= i <= n) of R. For a general ring R a Noetherian H-module M is comultiplication if and only if the R-P-module M-P is comultiplication for every maximal ideal P of R, but it is shown that this is not true in general. It is shown that comultiplication modules and quasi-injective modules are related in certain circumstances.