Exercise 5, § 13

Let K be a commutative field and L a finite set. Denote the algebra constructed in Exercise 17 of VIII, p. 169 by B, the central element $1-\sum_{\lambda}c_{\lambda}+\sum_{\lambda\not=\mu}r_{\lambda \mu}$ of B by $\varepsilon$, and the algebra $B/B\varepsilon$ by A. The K-algebra A is finite-dimensional, its radical $\mathfrak{r}$ has square zero, and $A/\mathfrak{r}$ is isomorphic to $K^L$. Prove that A is not isomorphic to a product of two nonzero algebras.