Exercise 8, § 12

Let K be a commutative field and L a radical extension of K of finite degree $>1$. Let V be an infinite-dimensional vector space over L, and let A be the K-subalgebra of End$_L(V)$ generated by the endomorphisms of finite rank and the identity mapping. Prove that the L-algebra $A_{(L)}$ has nonzero radical even though its center is equal to L (use Exercise 7).