  In this paper, we study the Noetherian dimension of sum of certain modules.
It is proved that for any module M which is an irredundant sum of submodules,
each of which has Noetherian dimension less than alpha, if M has finite
spanning dimension (fsd-module, for short) or it is a weakly atomic module,
then Noetherian dimension M less than alpha. Here, by a weakly atomic module we
mean a module M for which every proper non-small submodule N, has Noetherian
dimension strictly less than that of M. Also, it is proved that if M is an AB5*
module with Noetherian dimension and N_i is a family of submodules of M such
that Noetherian dimension M over N_i, less than alpha, for each i, then
Noetherian dimension M over intersection of N_is less than alpha. Using this,
we give a structure theorem for alpha-short modules in the category of AB5* and
finally, we classify alpha-short modules in this category.
