Some aspects of ( ! + 1 ) -projectives

In this paper, we prove that the (ω + 1 ) -projective (cid:3) -modules are direct sums of countably generated modules each of which has length at most (ω + 1 ) . We also give an example that this is not true for all other (ω + k ) -projectives with 2 ≤ k < ω . Certain related assertions are established as well. These results are then used to discuss the class of highly essentially finitely indecomposable modules