Exercise 23, § 8

Algebra, III, 643

In every A-algebra E, if $x_1, \ldots, x_n$ are elements of E, we write

[x_1x_2 \ldots x_n] = \sum_{\sigma \in S_n} \varepsilon_\sigma x_{\sigma(1)}x_{\sigma(2)} \ldots x_{\sigma(n)}.

Show that if E admits a finite generating system over A, there exists an integer N such that $[x_1x_2 \ldots x_N] = 0$ for all elements $x_j$ ( $1 \leq j \leq N$ ) of E.

Solution

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