Exercise 2, § 3

Let A be the Banach algebra of sequences $x= (x_n)_{n\geqslant 1}$ of complex numbers tending to 0, with $\|x\|=$ sup$_i|x_i|$. Let I be the set of elements of A of finite support. Then I is a dense ideal of A, and is not contained in any maximal ideal. (Observe that $A^2= A$, and use Exercise 1.)