Generators for the semimodule of varieties of a free module
Let \(R\) be a commutative ring with identity, and let \(M\) be an \(R\)-module. For a submodule \(N\) of \(M\) let \((N:M)= \{r\in R\mid rM \subseteq N\}\). The submodule \(N\) is called prime if \((N:M)\) is a prime ideal of \(R\) and the \(R/(N:M)\)-module \(M/N\) is torsion-free. In analogy to the spectrum of a ring one can define the set \(\text{Spec}(M)\) to be the collection of all prime submodules of \(M\). Let \(V(N)\), the variety of \(N\), be the collection of all prime submodules containing \(N\). The collection of all such varieties is denoted by \(\zeta(M)\). It can be shown that \(\zeta(M)\) is a semi-module over the semi-ring \(\zeta(R)\). This paper investigates the question when \(\zeta(M)\) is finitely generated as a semi-module over the semi-ring \(\zeta(R)\). The main result is that, if \(R\) is an infinite Noetherian UFD and \(M\) is a finitely generated \(R\)-module, then \(\zeta(M)\) is finitely generated if and only if \(M\cong R\oplus T\), where \(T\) is a finite \(R\)-module.
scientific article; zbMATH DE number 1455479
scientific article