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34 2. Basic notions of representation theory and “right A-module” are particular cases of the notion of bimodules namely, a left A-module is the same as an (A, k)-bimodule, and a right A-module is the same as a (k, A)-bimodule. Let B be a k-algebra, W a left B-module, and V a right B- module. We denote by V ⊗B W the k-vector space (V ⊗k W ) / vb ⊗ w − v ⊗ bw | v ∈ V, w ∈ W, b ∈ B . We denote the projection of a pure tensor v ⊗ w (with v ∈ V and w ∈ W ) onto the space V ⊗B W by v ⊗B w. (Note that this tensor product V ⊗B W is the one defined in Remark 2.11.4.) If, additionally, A is another k-algebra and if the right B-module structure on V is part of an (A, B)-bimodule structure, then V ⊗B W becomes a left A-module by a (v ⊗B w) = av ⊗B w for any a ∈ A, v ∈ V , and w ∈ W . Similarly, if C is another k-algebra, and if the left B-module structure on W is part of a (B, C)-bimodule structure, then V ⊗B W becomes a right C-module by (v ⊗B w) c = v ⊗B wc for any c ∈ C, v ∈ V , and w ∈ W . If V is an (A, B)-bimodule and W is a (B, C)-bimodule, then these two structures on V ⊗B W can be combined into one (A, C)- bimodule structure on V ⊗B W . (a) Let A, B, C, D be four algebras. Let V be an (A, B)- bimodule, W a (B, C)-bimodule, and X a (C, D)-bimodule. Prove that (V ⊗B W ) ⊗C X ∼ V ⊗B (W ⊗C X) as (A, D)-bimodules. The isomorphism (from left to right) is given by the formula (v ⊗B w) ⊗C x → v ⊗B (w ⊗C x) for all v ∈ V , w ∈ W , and x ∈ X. (b) If A, B, C are three algebras and if V is an (A, B)-bimodule and W an (A, C)-bimodule, then the vector space HomA (V, W ) (the space of all left A-linear homomorphisms from V to W ) canonically becomes a (B, C)-bimodule by setting (bf) (v) = f (vb) for all b ∈ B, f ∈ HomA (V, W ), and v ∈ V and setting (fc) (v) = f (v) c for all c ∈ C, f ∈ HomA (V, W ) and v ∈ V .