Maximal ideals in module categories and applications

We study the existence of maximal ideals in preadditive categories defining an order $⪯$ between objects, in such a way that if there do not exist maximal objects with respect to $⪯$, then there is no maximal ideal in the category. In our study, it is sometimes sufficient to restrict our attention to suitable subcategories. We give an example of a category $\mathbf C_F$ofmodulesoverarightnoetherianring$R$ in which there is a unique maximal ideal. The category ${\mathbf{C}}_F$ is related to an indecomposable injective module $F$, and the objects of ${\mathbf{C}}_F$ are the $R$-modules of finite $F$-rank.