Zero and inverse element in abelian group of extensions of modules

Question

Let $A,B$ be modules. An extension of $A$ by $B$ is a short exact sequence $$B\to E\to A$$. Let $Ext(A,B)$ be a set of equivalence classes of such modules $E$. The book said that this admit an abelian group structure. I would like to confirm is it true that the zero object is the split short exact sequence $B\to A\oplus B\to A$? I wanna prove this, but I am stuck at making an isomorphism between $E$ and $E\oplus A\oplus B$. And I have no clue about the inverse element of $B\to E\to A$: that is $B\to X\to A$ such that $E\oplus X$ is isomorphic to $A\oplus B$. Any help is appreciated.

Answer 1

For two extensions $$A\to E \to B$$ and $$A\to F \to B$$ we define the sum of them to be be the quotient of the pullback $X$ on the morphisms $$\pi_e: E\to B$$ $$\pi_f: F\to B$$ with the module $$N=\{\imath_e(a)\oplus-\imath_f(a):a\in A\}$$

Then the baer sum is $X/N$. Let $A\oplus B$ be given and $E$ an extension, then we have. $$X=\{(a\oplus b)\oplus e:\pi(a\oplus b)=\pi_e(e)\}$$ $$=\{a\oplus b\oplus e:b=\pi_e(e)\}$$ $$=\{(a\oplus b)\oplus e:b=\pi_e(e)\}$$ $$\cong\{a \oplus e\}$$ $$=A\oplus E$$ and for the submodule we have through a similar reasoning that $N\cong A\cong A\oplus 0$ and taking their quotient you get the rest. I recommend you look into MacLanes book on it.