  We study the existence of maximal ideals in preadditive categories defining
an order $\preceq$ between objects, in such a way that if there do not exist
maximal objects with respect to $\preceq$, 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 $\mathbf
C_F$ of modules over a right noetherian ring $R$ 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.
