Exercise 17, § 9

Let K be a commutative field and L a set. a) Prove that there exists a K-algebra A admitting a basis consisting of the unit element, a family $(c_{\lambda})_{\lambda\in L}$, and a family $(r_{\lambda \mu})_{(\lambda ,\mu)\in L\times L}$, with multiplication table

$c^2_{\lambda}=c_{\lambda},c_{\lambda}c_\mu=r_{\lambda \mu}$ if $\lambda \not=\mu$

where $\delta_{\mu\nu}$ is the Kronecker delta function. Prove that $\mathfrak{R}(A)$ is the subalgebra of A generated by the $r_{\lambda \mu}$; we have $\mathfrak{R}(A)^2= 0$. b) Suppose Card(L) $\geqslant 2$. Prove that the center Z of A is the K-vector space generated by 1, the $r_{\lambda \lambda}$ for $\lambda \in L$, and, if L is finite, the element $\sum_{\lambda}c_{\lambda}-\sum_{\mu\not=\nu}r_{\mu\nu}$. Deduce that for $\lambda \in L$, there are no idempotents in Z congruent to $c_{\lambda}$ (mod $\mathfrak{R}(A)$).