Definition 2

Let $A$ be a ring. An $A$-algebra $R$ (not necessarily commutative) is called integral over $A$ if every element of $R$ is integral over $A$. $R$ is called finite over $A$ if $R$ is a finitely generated $A$-module.

It follows from Proposition 1 that every finite $A$-algebra is integral; if $R$ is commutative and a finite $A$-algebra, $R$ is obviously a finitely generated $A$-algebra; the converse is false.

Example (4) If $M$ is a finitely generated $A$-module, the algebra $\mathrm{End}_A(M)$ of endomorphisms of $M$ is integral over $A$ by virtue of (E_{III}); in particular, for every integer $n$, the matrix algebra $M_n(A) = \mathrm{End}_A(A^n)$ is integral (and even finite) over $A$.