Bourbaki

Exercise 7, § 5

Algebra, II, 399

Let ρ:A→B\rho : A \to B be a ring homomorphism. For every left B-module F, define a canonical B-module homomorphism

F→ρ~(ρ∗(F)) F \to \tilde{\rho}(\rho_*(F))
and show that if E is a left A-module, the composite mappings
ρ~(E)→ρ~(ρ∗(ρ~(E)))→ρ~(E) \tilde{\rho}(E) \to \tilde{\rho}(\rho_*(\tilde{\rho}(E))) \to \tilde{\rho}(E)
ρ∗(F)→ρ∗(ρ~(ρ∗(F)))→ρ∗(F) \rho_*(F) \to \rho_*(\tilde{\rho}(\rho_*(F))) \to \rho_*(F)
are the identity mappings.

Solution

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Transcribed from the printed volume. Not translated.