Modules with semi-local endomorphism ring

Abstract

We use the concept of dual Goldie dimension and a characteriza- tion of semi-local rings due to Camps and Dicks (1993) to find some classes of modules with semi-local endomorphism ring. We deduce that linearly com- pact modules have semi-local endomorphism ring, cancel from direct sums and satisfy the n th root uniqueness property. We also deduce that modules over commutative rings satisfying AB5* also cancel from direct sums and satisfy the n th root uniqueness property. Let R be an associative ring with 1 and let Af be a right unital i?-module. A finite set Ax, ... , An of proper submodules of M is said to be coindependent if for each /, 1 < i < n, Ai + f\j:jii Aj = M, and a family of submodules of M is said to be coindependent if each of its finite subfamilies is coindependent. The module M is said to have finite dual Goldie dimension if every coindependent family of submodules of M is finite. It can be shown that, in this case, there is a maximal coindependent family of submodules of Af. If this set is finite, then its cardinality (denoted by codim(Af) ) is uniquely determined and is called the dual Goldie dimension of Af. If this set is infinite we set codim(Af ) = oo and say that Af has infinite dual Goldie dimension. A module with dual Goldie dimension 1 is said to be hollow, and a cyclic hollow module is said to be local. We have