Corollary 2

Let $A$ be a ring, $R$ an A-algebra (not necessarily commutative) and $E$ a set $\&$ elements $\& R$ which are pairwise permutable and integral over $A$. Then the sub-A-algebra $B \& R$ generated by $E$ is integral over $A$.

Every element of $B$ belongs to a sub-A-algebra of $B$ generated by a finite subset of $E$.

Remark (3) It follows from Proposition 4 that every commutative A-algebra integral over $A$ is the union of a right directed family of finite subalgebras over $A$.