Exercise 6, § 5

Let K be a commutative field and E, F two K-algebras each with a unit element which is identified with the unit element 1 of K. Consider the K-algebra $T(E \oplus F)$, where $E \oplus F$ is considered as a vector space over K; E (resp. F) is identified with a vector subspace of $T(E \oplus F)$ under the canonical injection. Let $J$ be the two-sided ideal of $T(E \oplus F)$ generated by the elements $x_1 \otimes x_2 - (x_1x_2)$ and $y_1 \otimes y_2 - (y_1y_2)$, where $x_i \in E$, $y_i \in F$, $i = 1, 2$; let $E * F$ denote the quotient algebra $T(E \oplus F)/J$; it is called the free product of the K-algebras E and F; let $\phi$ denote the canonical homomorphism of $T(E \oplus F)$ onto $E * F$ and $\alpha$ and $\beta$ the restrictions of $\phi$ to E and F; $\alpha$ and $\beta$ are K-*algebra* homomorphisms.

(a) Show that for every ordered pair of K-homomorphisms $u : E \to G,$ v : F → G into a K-algebra G, there exists one and only one K-homomorphism w : E \* F → G such that u = w ∘ α and v = w ∘ β (which justifies the terminology).

(b) Let R_n be the vector subspace of T(E ⊕ F) the sum of the T^k(E ⊕ F) for 0 ≤ k ≤ n and write J_n = J ∩ R_n. Show that, if we consider a basis of E consisting of 1 and a family (e_\lambda)_{\lambda \in L} and a basis of F consisting of 1 and a family (f_\mu)_{\mu \in M}, then a supplementary subspace of J_n + R_{n-1} is obtained in R_n by considering the subspace of R_n with basis the elements of the form z_1 \otimes z_2 \otimes \cdots \otimes z_n where, either z_{2j+1} is equal to one of the e_\lambda for 2j + 1 ≤ n and z_{2j} is equal to one of the f_\mu for 2j ≤ n, or z_{2j+1} is equal to one of the f_\mu for 2j + 1 ≤ n and z_{2j} is equal to one of the e_\lambda for 2j ≤ n (argue by induction on n).

(c) Let P_n = \phi(R_n) \subset E \* F; then P_n \subset P_{n+1}, P_0 = K, P_h P_k \subset P_{h+k} and E \* F is the union of the P_n. Show that there is a canonical vector space isomorphism, for n ≥ 1,

where we write G_k = E/K for k odd, G_k = F/K for k even. In particular the homomorphisms α and β are injective, which allows us to identify E and F with sub-K-algebras of E \* F. P'_n (resp. P''_n) is used to denote the inverse images in P of G_1 \otimes G_2 \otimes \cdots \otimes G_n (resp. G_2 \otimes G_3 \otimes \cdots \otimes G_{n+1}); the height of an element z \in E \* F, denoted by h(z), is the smallest integer such that z \in P_n; if h(z) = n and z \in P'_n (resp. z \in P''_n), z is called pure odd (resp. pure even). Then P_n = P'_n + P''_n and P'_n \cap P''_n = P_{n-1} for n ≥ 1.

(d) Show that, if u, v are two elements of E \* F such that h(u) = r, h(v) = s, then h(uv) ≤ h(u) + h(v) and that h(uv) = h(u) + h(v) unless u and v are pure and, either r is even and u, v are not of the same parity, or r is odd and u, v are of the same parity. (If for example r is even, u \in P'_r, v \in P'_s, show that uv \in P_{r+s-1} is possible only if u \in P_{r-1} or v \in P_{s-1}.)

(e) Suppose that E has no divisors of zero and, in the notation of (d), suppose that r is even, u pure even and v pure odd. Show that h(uv) = r + s - 1, unless there exist two invertible elements x, y of E such that uy \in P_{r-1} and xb \in P_{s-1}. (Let (a_i) (resp. (b_j)) be the basis of the supplementary subspace of J_r + R_{r-1} in R_r (resp. of J_s + R_{s-1} in R_s) considered in (b); write u = \sum_i \phi(a_i)x_i and v = \sum_j y_j\phi(b_j), where x_i and y_j are in E, and show that, if uv \in P_{r+s-2}, then necessarily x_iy_j \in K.) Treat similarly the other cases where h(uv) < h(u) + h(v) when E and F are assumed to have no divisors of zero.

(f) Deduce from (d) and (e) that, when E and F have no divisors of zero, nor does E \* F.

(g) Generalize the above definitions and results to any finite number of unital K-algebras.