Maximal ideals in module categories and applications

TL;DR

This paper investigates the existence of maximal ideals in certain module categories, linking their existence to the presence of maximal objects under a defined order, and provides an example with a unique maximal ideal.

Contribution

It introduces a method to determine maximal ideals in module categories via object ordering and constructs an example with a unique maximal ideal in a specific subcategory.

Findings

Existence of maximal ideals correlates with maximal objects under a category order.

A specific subcategory of modules over a noetherian ring has a unique maximal ideal.

The approach simplifies the study of maximal ideals by focusing on suitable subcategories.

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.