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