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.

Let R be an integral domain, and let A, A(1), A(2), ..., A s be overrings of R, where A is of the form S-1 R, where S = R \ p1 boolean OR ... boolean OR p(n) for for some prime ideals p(i), and where each A(i), i >= 2, is of the form S-i(-1) R for some multiplicatively closed subset S-i of R. It is shown that if A subset of A(1) boolean OR ... boolean OR A(s), then A subset of A(i) for some i.

A prime ideal p is said to be strongly prime if whenever p contains an intersection of ideals, p contains one of the ideals in the intersection. A commutative ring with this property for every prime ideal is called strongly zero-dimensional. Some equivalent conditions are given and it is proved that a zero-dimensional ring is strongly zero-dimensional if and only if the ring is quasi-semi-local. A ring is called strongly n-regular if in each ideal a, there is an element a such that x=ax for all x ∈ an. Connections between the concepts strongly zero-dimensional and strongly n-regular are considered.

This paper is concerned with finite unions of ideals and modules. The first main result is that, if N ⊆ N ₁ ∪N ₂ ∪ … ∪ N _s is a covering of a module N by submodules N _i , such that all but two of the N _i are intersections of strongly irreducible modules, then N ⊆ N _k for some k. The special case when N is a multiplication module is considered. The second main result generalizes earlier results on coverings by primary submodules. In the last section unions of cosets is studied.

Three related properties of a module are investigated in this article, namely the Nakayama property, the Maximal property, and the S-property. A module M has the Nakayamapropertyif aM=M for an ideal a implies that sM=0 for some s∈a+1. A module M has the Maximal property if there is in M a maximal proper submodule, and finally, M is said to have the S-property if S^{−1}M = 0 for a multiplicatively closed set S implies that sM=0 for some s∈S.