Let $0 \rightarrow M \rightarrow E \rightarrow K \rightarrow 0$ and $0 \rightarrow M \rightarrow E' \rightarrow K' \rightarrow 0$ be short exact sequences with $M$ a left $R$-module and $E, E'$ injective left R-modules. Prove the dual version of Schanuel’s Lemma by showing that E ⊕ K′ $\cong$ E′ ⊕ K.
Two questions:
- What is the dual of Schanuel's Lemma precisely?
- How does "showing that E ⊕ K′ $\cong$ E′ ⊕ K" succeed in proving the dual of Schanuel's Lemma?