Exercise 1, § 3

Let E be an associative commutative and unital algebra over a commutative field K ; let x = (x_i)_{i \in I}, be a family of pairwise permutable elements of E. Let U be the subring of K(X_i)_{i \in I} consisting of all f \in K(X_i)_{i \in I} such that x is substitutable in f. Show that if in E the set of non-invertible elements is an ideal, then the same is true of U. Show that the ideal of non-invertible elements in U is then maximal.