Similar submodules of projective modules

TL;DR

This paper introduces a new similarity relation for submodules of modules over rings, establishing bounds on maximal submodules and exploring structural properties of projective modules, with applications to matrix rings.

Contribution

It extends classical similarity notions to submodules, providing bounds on maximal submodules and characterizing projective modules with finite length.

Findings

Lower bound for the number of maximal submodules in projective modules.

Constructs a canonical map from maximal submodules to maximal right ideals.

Shows that certain projective modules decompose into local summands when endomorphism rings are Artinian.

Abstract

We introduce a similarity relation between submodules of a module $M$ over a ring $R$, extending the classical notion of similarity for right ideals. Focusing on (faithfully) projective modules, we establish a sharp lower bound for the number of maximal submodules: if $N$ is a maximal submodule of $M$, then either $N$ is fully invariant or $N$ is similar to at least $1+|S|$ distinct maximal submodules, where $S$ is the eigenring of $N$; in particular, $|{\rm Max}(M)|\geq 1+|S|\geq 3$ in the latter case. For projective modules, we construct a canonical one-to-one map from ${\rm Max}(M)$ into ${\rm Max}_r({\rm End}_R(M))$. When $M$ is faithfully projective and ${\rm End}_R(M)$ is right Artinian, we prove that $M$ has finite length and decomposes into a direct sum of local summands. Conversely, if $M$ is a projective right $R$-module with finite length, then $E_E$ has finite length with $\ell(E_E)\leq \ell(M_R)$; moreover, if $M$ is a faithfully projective $R$-module, then $\ell(E_E)=\ell(M_R)$; conversely, if $\ell(E_E)=\ell(M_R)$ holds, then $M$ is slightly compressible. These results are applied to obtain lower bounds on the number of maximal one-sided ideals that are not two-sided, with explicit consequences for matrix rings over infinite algebras.