THE LENGTHS OF ARTINIAN MODULES WITH COUNTABLE NOETHERIAN DIMENSIONS

It is shown that if M is an Artinian module over a ring R, then M has Noetherian dimension alpha, where alpha is a countable ordinal number, if and only if omega(alpha) + 2 <= l(M) <= omega(alpha+1), where l(M) is the length of M, i.e., the least ordinal number such that the interval [0, l(M)) cannot be embedded in the lattice of all submodules of M.