Christian Gottlieb

On finite unions of ideals and cosets

One only needs little experience from commutative algebra to perceive that whereas any intersection of ideals again is an ideal it is quite rare that a finite union of ideals itself is an ideal. This fact is mostly used when the purpose is to, within a given ideal I, find an element which avoids (i.e. lies outside) a finite union I1 ∪ I2 ∪ · · · ∪ In of ideals. Usually much is achieved if the ideals can be avoided each at a time. Indeed we all know ”the prime avoidance lemma”:

An integer-valued function related to the number of generators of modules over local rings of small dimension

An integer-valued function related to the number of generators of modules over local rings of small dimension

The Nakayama Property of a Module and Related Concepts

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 Nakayama property if 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.

Finite unions of submodules

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.