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.

Take any square matrix and compute its characteristic polynomial. Then substitute the matrix for the variable and start computing. This is a good exercise for a beginner working with matrices. Moreover, the result is quite remarkable. It is, namely, always the zero matrix.

It would require an extraordinary intuition to suspect this, by no means, intuitive result. So this is one of many examples where mathematics surprises us in a most beautiful way. This is the so called Cayley-Hamilton theorem. We hope it will be of interest for a student, who has tried this in some examples, to see a proof of the theorem. Usually the theorem is proved in more advanced textbooks, with techniques not found in a first course of linear algebra. Here we give two elementary proofs, the first rather abstract, the second more concrete.

Let R be an integral domain, and let A,A₁,A₂,…,A_s be overrings of R, where A is of the form S⁻¹R, where S=R∖𝖕₁∪⋯∪𝖕_n for some prime ideals 𝖕_i, and where each A_i, i≥2, is of the form for some multiplicatively closed subset S_i of R. It is shown that if A⊆A₁∪⋯∪A_s, then A⊆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.