Exercise 4, § 9

Let A be an algebra over a commutative field K and $\mathfrak{a}$ a two-sided ideal of A. Let B be the subalgebra of A generated by 1 and $\mathfrak{a}$. Prove that the radical of B is contained in that of A (observe that if S is a simple A-module, then we have either $\mathfrak{a}x= 0$ for every $x\in S$ or $\mathfrak{a}x= S$ for every $x\not= 0$ in S).