If R,S are rings, A_R, _RB_s, _sC are (bi)modules and D an abelian group, define a middle linear map to be a function ∫: A X B X C → D such that

(i) ∫(a + a',b,c) = ∫(a,b,c) +∫(a',b,c);

(ii) ∫(a,b + b',c) = ∫(a,b,c) + ∫(a,b',c);

(iii) ∫(a,b,c + c') = ∫(a,b,c) + ∫(a,b,c');

(iv) ∫(ar,b,c) = ∫(a,rb,c) for r ϵ R;

(v) ∫(a,bs,c) = ∫(a,b,sc) for s ϵ S.

(a) The map i: A X B X C → (A ⊗_R B) ⊗_s C given by (a,b,c) |→ (a⊗b)⊗ c is middle linear.

(b) The middle linear map i is universal; that is, given a middle linear map g: A X B X C → D, there exists a unique group homomorphism g̅: (A ⊗_RB) ⊗_s C → D such that g̅i = g.

(c) The map j : A X B X C → A ⊗_R (B ⊗_s C). given by (a,b,c) |→ a ⊗ (b⊗c) is also a universal middle linear map.

(d) (A⊗_R B) ⊗_S C ≅ A⊗_R (B⊗_s C) by (b), (c), and Theorem 1.7.10.

(e) Define a middle linear function on n (bi)modules (n ≥ 4) in the obvious way and sketch a proof of the extension of the above results to the case of n (bi)- modules (over n - 1 rings).

(f) If R = S, R is commutative and A,B,C,D are R-modules, define a trilinear map A X B X C → D and extend the results of (a),(b),(c) to such maps.