Exercise 21, § 9

Let A be a ring, $u: P\rightarrow M$ a surjective homomorphism of A-modules, and N its kernel. We denote by End$_A(P; N)$ (resp. Aut$_A(P; N)$) the ring of endomorphisms (resp. the group of automorphisms) $u$ of P such that $u(N)\subset N$.

a) Define an exact sequence

$0\rightarrow$ Hom$_A(P,N)\rightarrow^i$ End$_A(P; N)\rightarrow^p$ End$_A(M)$

where the image of $i$ is a two-sided ideal $\mathfrak{I}$ of the ring End$_A(P; N)$.

b) Suppose that $(P, u)$ is a projective cover of M. Then $p$ is surjective and induces a surjective homomorphism $p':$ Aut$_A(P; N)\rightarrow$ Aut$_A(M)$. We have an exact sequence of groups

$0\rightarrow 1_P+\mathfrak{I}\rightarrow$ Aut$_A(P; N)\rightarrow^{p'}$ Aut$_A(M)\rightarrow 0$;

if P is free and N is not reduced to 0, then the homomorphism $p'$ is not injective.