Abstract
All rings are commutative. Let $M$ be a module. We introduce the property $({\bf P})$: Every endomorphism of $M$ has a non-trivial invariant submodule. We determine the structure of all vector spaces having $({\bf P})$ over any field and all semisimple modules satisfying $({\bf P})$ over any ring. Also, we provide a structure theorem for abelian groups having this property. We conclude the paper by characterizing the class of rings for which every module satisfies $({\bf P})$ as that of the rings $R$ for which $R/\mathfrak{m}$ is an algebraically closed field for every maximal ideal $\mathfrak{m}$ of $R$.
Highlights
The notion studied in this article has its roots in operator theory
Following [1], the research on this problem was initiated by J. von Neumann who proved in the early thirties of the last century that every linear compact operator on a Hilbert space has a non-trivial invariant closed subspace
Example 2.3. (i) It is clear that every module having a non-trivial fully invariant submodule has (P)
Summary
The notion studied in this article has its roots in operator theory. Let B(H) be the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H. (i) It is clear that every module having a non-trivial fully invariant submodule has (P). The following are equivalent: (i) Every K-vector space of dimension t (2 ≤ t ≤ n) has (P); (ii) Every monic polynomial P (X) ∈ K[X] of degree t (2 ≤ t ≤ n) has a root in the field K. Let M be an infinitely generated semisimple module.
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