# max-projective modules

## TL;DR

This paper characterizes right almost-$QF$ and max-$QF$ rings, showing their equivalence under certain conditions and providing structural descriptions for these classes of rings.

## Contribution

It introduces the concepts of max-projective modules and right max-$QF$ rings, and characterizes these rings in terms of their structural decomposition and properties.

## Key findings

- Right Hereditary right Noetherian rings are right max-$QF$ iff they are right almost-$QF$.
- A right Hereditary ring is max-$QF$ iff every injective simple module is projective.
- A commutative Noetherian ring is almost-$QF$ iff it decomposes into a product of a $QF$ ring and a small ring.

## Abstract

A right $R$-module $M$ is called max-projective provided that each homomorphism $f:M \to R/I$ where $I$ is any maximal right ideal, factors through the canonical projection $\pi : R \to R/I$. We call a ring $R$ right almost-$QF$ (resp. right max-$QF$) if every injective right $R$-module is $R$-projective (resp. max-projective). This paper attempts to understand the class of right almost-$QF$ (resp. right max-$QF$) rings. Among other results, we prove that a right Hereditary right Noetherian ring $R$ is right almost-$QF$ if and only if $R$ is right max-$QF$ if and only if $R=S\times T$ , where $S$ is semisimple Artinian and $T$ is right small. A right Hereditary ring is max-$QF$ if and only if every injective simple right $R$-module is projective. Furthermore, a commutative Noetherian ring $R$ is almost-$QF$ if and only if $R$ is max-$QF$ if and only if $R=A \times B$, where $A$ is $QF$ and $B$ is a small ring.