2025-10-27

This following is a very brief summary of what happened in class on 2025-10-27.

After recapping the situation of the functor ${\mathrm{Hom}}_R(M,-):R\mathbf{\text{-Mod}}\to \mathbf{\text{Ab}}$ and projective modules, we repeated the analysis (skipping most proofs) for the functor ${\mathrm{Hom}}_R(-,N):(R\mathbf{\text{-Mod}}{)}^{\text{op}}\to \mathbf{\text{Ab}}$.

We noted the following:

• The functor ${\mathrm{Hom}}_R(-,N)$ is left exact. Modules $I$ for which this functor is exact are called injective.
• A non-obvious fact is that a module $I$ is injective if and only if it has the following property: whenever $I$ is a submodule of an $R$-module $M$, it is (isomorphic to) a direct summand of $M$.
• We noted (without proof) some examples of injective modules, and some examples of non-injective modules.

Concepts

• Exact Sequences IV - Exact Sequences and Functors

References

• Dummit & Foote, Abstract Algebra: Section 10.5