Exercise 16, § 9

Let A be a ring and $\mathfrak{a}$ a two-sided ideal contained in the radical of A. Let M and N be A-modules and $u: M\rightarrow N$ a homomorphism. We denote by $\overline{u}: M/\mathfrak{a}M\rightarrow N/\mathfrak{a}N$ the homomorphism of $A/\mathfrak{a}$-modules deduced from $u$.

a) Suppose that M and N are finitely generated and N is projective. If $\overline{u}$ is bijective, then so is $u$.

b) Give an example with M and N free of rank 1 and $\overline{u}$ injective but Ker($u$)$\not= 0$. c) Suppose that M and N are projective. If $\overline{u}$ induces an isomorphism to a direct factor, then $u$ is injective (reduce to the case when M and N are free and equal and $\overline{u}$ is the identity, and then use a)). If, moreover, M is finitely generated, then the image of $u$ is a direct factor of N. d) Suppose that M is finitely generated and N is projective. If $u$ is bijective, then so is $u$ (apply c) to a homomorphism $v: N\rightarrow M$ such that $v=u^{-1}$ to conclude that N is finitely generated). e) If M is projective and $M/\mathfrak{a}M$ is zero, then M is zero.