On double chains and inverse double chains of modules over a commutative ring

We consider infinite double chains ⋯ 𝑁−1 ⊂ 𝑁0 ⊂ 𝑁1 ⊂ ⋯ and infinite inverse double chains 𝑁0 ⊂ 𝑁1 ⊂ 𝑁2 ⊂ ⋯⁢⋯ ⊂ 𝑀2 ⊂ 𝑀1 ⊂ 𝑀0 of submodules of a module M over a commutative ring R. We shall give conditions under which such chains exist. Examples show that they do not always exist even though M 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.