Exercise 4, § 1

Commutative Algebra, I, 39

Consider a commutative diagram of commutative groups

\begin{array}{ccccccc} A & \longrightarrow & B & \longrightarrow & C & \longrightarrow & D \\ a \downarrow & & b \downarrow & & c \downarrow & & d \downarrow \\ A' & \longrightarrow & B' & \longrightarrow & C' & \longrightarrow & D' \end{array}

where the two rows are assumed to be exact.

(a) Show that, if $a$ is surjective and $b$ and $d$ injective, then $c$ is injective. (b) Show that, if $d$ is injective and $a$ and $c$ surjective, then $b$ is surjective.

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