Introduction

Injective modules are a cornerstone of module theory, serving as a bridge between homological algebra and ring theory. In the context of module over a ring, an injective module is a module that satisfies the property that the hom functor $\text{Hom}(-, M)$ is exact. This property, known as the "exactness of the hom functor," ensures that injective modules preserve certain structural properties under homomorphisms, making them indispensable in the study of module categories. The exploration of injective modules encompasses their definition, equivalence conditions, and concrete examples, all of which are critical for understanding their role in advanced algebraic structures. This article delves into these aspects, emphasizing the interplay between injective modules and the broader landscape of ring theory.

Definition

An injective module over a ring $ R $ is a module $ M $ such that for every injective homomorphism $ f: A \to B $ of $ R $-modules, the induced map $ \text{Hom}(B, M) \to \text{Hom}(A, M) $ is also exact. Equivalently, $ M $ is injective if and only if the functor $ \text{Hom}(-, M) $ is exact. This definition is particularly significant in the context of left $ R $-modules, where injectivity often relates to the module's ability to "absorb" homomorphisms. For example, over a field $ \mathbb{F} $, injective modules are precisely divisible modules, as the hom functor preserves exactness when applied to free modules. However, in more general rings, injective modules can exhibit complex behavior, necessitating the study of their structural properties.

Equivalence Conditions

The injectivity of a module is closely tied to its relationship with other module properties, such as flatness and torsion-freeness.