We consider infinite double chains β― πβ1 β π0 β π1 β β― and infinite inverse double chains π0 β π1 β π2 β β―β’β― β π2 β π1 β π0 of submodules of a module MΒ over a commutative ringΒ R. We shall give conditions under which such chains exist. Examples show that they do not always exist even thoughΒ MΒ satisfies neither the ascending nor the descending chain conditions. The concepts of m-minimal and m-maximal submodules will be introduced as main tools. Also we shall give conditions under which a Noetherian module is Artinian and conditions under which an Artinian module is Noetherian.