Corollary 1

Let $A$ be a ring and $R$ a (Commutative)A-algebra. The set of elements $\& R$ integral over $A$ is a subalgebra of $R$.

In fact, if $x, y$ are two elements of $R$ integral over $A$, it follows from Proposition 4 that $A[x, y]$ is a finitely generated A-module; as it contains $x + y$ and $xy$, the corollary follows from Proposition 1.

In a non-commutative algebra the sum and product of two integral elements over $A$ are not necessarily integral over $A$ (Exercise 4).