Similar submodules of projective modules

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.