The length of Artinian modules with countable Noetherian dimension

‎It is shown that‎ ‎if M is an Artinian module over a ring‎ ‎R‎, ‎then M has Noetherian dimension α‎, ‎where α is a countable ordinal number‎, ‎if and only if ωα+2≤l(M)≤ωα‎+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.