Exercise 11, § 6

a) A topological ring with an identity element is said to be a Gelfand ring if $A^*$ is open in $A$ and if the topology induced on $A^*$ by the topology of $A$ is compatible with the multiplicative group structure of $A^*$. Every Hausdorff topological division ring is a Gelfand ring [cf. Exercise 20 e)].

b) Show that if $A$ is a Gelfand ring then so is every matrix ring $M_n(A)$ endowed with the product topology (on $A^{n^2}$). (Proof by induction on $n$).