Exercise 4, § 5

Algebra, II, 398

Let $\rho : A \to B$ be a ring homomorphism. For every A-module E, show that the diagram

\begin{array}{ccccc} E & \xrightarrow{c_E} & E^{**} & \longrightarrow & (E^{**})_{(B)} \\ \downarrow & & \downarrow & & \downarrow \nu_{E^*} \\ E_{(B)} & \xrightarrow{c_{E_{(B)}}} & (E_{(B)})^{**} & \xrightarrow{\iota \nu_E} & ((E^*)_{(B)})^* \end{array}

is commutative.

Solution

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