The Hyperradical and the Hopkins–levitzki Theorem for Modular Lattices

Many arguments in the Theory of Rings and Modules are, on close inspection, purely Lattice theoretic arguments. Cǎlagǎreanu has a long repertoire of such results in his book. The Hopkins-Levitzki Theorem is interesting from this point of view, because a special case of it lends to an obvious lattice theory approach, but the rest is a little more subtle. Albu and Smith have obtained some sufficient conditions for the question of when Artinian implies Noetherian. Here we present a new approach, using the concept of Hyperradical; we obtain necessary and sufficient conditions.