$\Gamma_{\mathfrak a}(I)$ is an injective $R$-module for every injective $R$-module $I$

Question

Is there a proof for Proposition 2.1.4 of Local Cohomology book by Brodmann-Sharp not using Artin–Rees Lemma?

Proposition 2.1.4: Let $I$ be an injective $R$-module. Then $\Gamma_{\mathfrak a}(I)$ is also an injective $R$-module.

Answer 1

There is one at exercise 10.1.11 of the book, which I gave a proof (not in details):

$I= \oplus E(R/\mathfrak p)$ being injective we have that $$\Gamma_{\mathfrak a}(I) = \Gamma_{\mathfrak a}({\oplus E(R/\mathfrak p)} )= \oplus \Gamma_{\mathfrak a}({E(R/\mathfrak p)})=\oplus_{\mathfrak a\subseteq \mathfrak p} E(R/\mathfrak p)$$ is injective, since $R$ is noetherian (see Theorem 18.4 of Matsumura's Commutative Ring Theory).