- #1

- 45

- 1

**Given:**the short exact sequences 0 → M → E → K → 0 and 0 → M → E' → K' → 0 where M is a left R-module and E and E' are injective left R-modules.

**Prove:**E

**⊕**K' ≅ E'

**⊕**K.

First, let

*f*be the morphism represented by M → E and

*g*be the morphism represented by M → E'. Therefore we can construct a pushforward which is the object X together with the morphism i

_{1}: E → X and i

_{2}: E' → X

How do I proceed from there? I need to then somehow chain together the morphisms from this pushout to get two short exact sequences, show that they split, and then show that they are equal each to the direct sum.