Exercise 6, § 1

Commutative Algebra, I, 40

Suppose that an exact sequence $0 \to A \xrightarrow{u} A \xrightarrow{v} A''$ and two injective homomorphisms $A' \xrightarrow{a'} C',\ A'' \xrightarrow{a''} C''$ are given, where $A,\ A',\ A'',\ C',\ C''$ are modules over the same ring. Show that, if $C'$ is an injective module (Algebra, Chapter II, § 2, Exercise 11), there exists an injective homomorphism $a : A \to C' \oplus C''$ such that the diagram

\begin{array}{ccc} A' \xrightarrow{u} & & A \xrightarrow{v} A'' \\ a' \downarrow & & a \downarrow \quad a'' \downarrow \\ C' \xrightarrow{i} & & C' \oplus C'' \xrightarrow{p} C'' \end{array}

are commutative ( $i$ and $p$ being the canonical mappings).

Solution

No solution has been written yet.