Abstract
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.
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