Direct Sums of Special Modules (10.5.3-5)

Dummit and Foote Abstract Algebra, section 10.5, exercise 4:

3. Prove $Q_1 \oplus Q_2$ is a projective $R$-module if and only if $Q_1$ and $Q_2$ are projective $R$-modules.
4. Prove $Q_1 \oplus Q_2$ is an injective $R$-module if and only if $Q_1$ and $Q_2$ are injective $R$-modules.
5. Prove $Q_1 \oplus Q_2$ is a flat $R$-module if and only if $Q_1$ and $Q_2$ are flat $R$-modules. Prove $\sum A_i$ is a flat $R$-module if and only if each $A_i$ is a flat module.

Proof: Lemma: Let $\mathcal{F}_i$ be functors of $R$-modules. $\bigoplus \mathcal{F}_i$ is exact if and only if every $\mathcal{F}_i$ is exact. Proof: ($\Leftarrow$) Letting $0 → L → M → N → 0$ be exact by $ψ$ and $φ$, since $\mathcal{F}_i$ are exact functors, then $(\bigoplus \mathcal{F}_i)(ψ)$ is injective by observation of components, likewise $(\bigoplus \mathcal{F}_i)(φ)$ is surjective, and $(\bigoplus \mathcal{F}_i)(φ)$ is zero on and only on elements whose individual coordinates belong to the images of their respective functored $ψ$ homomorphisms, i.e. $\text{ker }(\bigoplus \mathcal{F}_i)(φ)=\text{img }(\bigoplus \mathcal{F}_i)(ψ)$. ($⇒$) For some inexact $\mathcal{F}_n$, we can observe inexactness in $(\bigoplus \mathcal{F}_i)$ in a natural fashion. For example, if $\text{ker }(\mathcal{F}_n)(φ)≠\text{img }(\mathcal{F}_n)(ψ)$, then the kernel and images of $(\bigoplus \mathcal{F}_i)(φ)$ and $(\bigoplus \mathcal{F}_i)(ψ)$ don't coincide by adducing the kernel-image inexactness in the coordinate in question.$~\square$

Proof: (3-5) $M_i$ are all projective (or injective or flat [here $I$ possibly infinite]) if and only if $Hom_R(M_i,\_)$ (or $Hom_R(\_,M_i)$ or $M_i \otimes_R \_$) are all exact if and only if $\bigoplus Hom_R(M_i,\_)≅Hom_R(\bigoplus M_i,\_)$ (or $\bigoplus Hom_R(\_,M_i)≅Hom_R(\_,\bigoplus M_i)$ or $\bigoplus (M_i \otimes \_) ≅ (\bigoplus M_i) \otimes \_)$) if and only if $\bigoplus M_i$ is projective (or injective or flat).$~\square$