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The functor you mention is faithful if and only if the functor $-\bigotimes -\bigotimes_A B :A-mod\to A-mod$ is faithful, ie iff $B$ is a faithfully flat faithful $A$-module. For a concrete counterexample take $f:\mathbb{Z}\to \mathbb{Q}$ and like Graham says this kills the torsion stuff. |
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The functor you mention is faithful if and only if the functor $-\bigotimes B :A-mod\to A-mod$ is faithful, ie iff $B$ is a faithfully flat $A$-module. For a concrete counterexample take $f:\mathbb{Z}\to \mathbb{Q}$ and like Graham says this kills the torsion stuff. |
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