integral

Let $B$ be a ring with a subring $A$. We will assume that $A$ is contained in the center of $B$ (in particular, $A$ is commutative). An element $x\in B$ is integral over $A$ if there exist elements $a_0,\mathrm{\dots },a_{n-1}\in A$ such that

$$x^n+a_{n-1}{x}^{n-1}+\mathrm{\cdots }+a_{1}x+a_0=0.$$

The ring $B$ is integral over $A$ if every element of $B$ is integral over $A$.