Exercise 1, § 3

Algebra, II, 395

Let $A$ be a ring, $E$ a right $A$ -module and $F$ a left $A$ -module. Let $f$ be a function defined on the set $S$ of all finite sequences

((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)) \quad (n \text{ arbitrary})

of elements of $E \times F$ with values in a set $G$ and such that:

(1)

f((x_1, y_1), \ldots, (x_n, y_n)) = f((x_{\sigma(1)}, y_{\sigma(1)}), \ldots, (x_{\sigma(n)}, y_{\sigma(n)}))

for every permutation

\sigma \in \mathfrak{S}_n

;

(2)

f((x_1 + x'_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)) \\ = f((x_1, y_1), (x'_1, y_1), (x_2, y_2), \ldots, (x_n, y_n));

(3)

f((x_1, y_1 + y'_1), (x_2, y_2), \ldots, (x_n, y_n)) \\ = f((x_1, y_1), (x_1, y'_1), (x_2, y_2), \ldots, (x_n, y_n));

(4)

f((x_1 \lambda, y_1), \ldots, (x_n, y_n)) = f((x_1, \lambda y_1), \ldots, (x_n, y_n))

for all

\lambda \in A

Show that there exists one and only one mapping $g$ of $E \otimes_A F$ into $G$ such that $f((x_1, y_1), \ldots, (x_n, y_n)) = g \left( \sum_{i=1}^n x_i \otimes y_i \right)$ . (Note that if

\sum_i x_i \otimes y_i = \sum_j x'_j \otimes y'_j,

the difference $\sum_i (x_i, y_i) - \sum_j (x'_j, y'_j)$ in the $\mathbf{Z}$ -module $\mathbf{Z}^{(E \times F)}$ is a linear combination with integer coefficients of elements of the form (2) of no. 1.)

Solution

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