General Ring Theory/Integral elements and extensions

Definition (integral element):

Let $R$ be a ring, and let $S/R$ be a ring extension. An element $s\in S$ is called integral over $R$ if there exist elements $a_{0},a_{1},\ldots ,a_{n-1}\in R$ such that

$s^{n}+a_{n-1}s^{n-1}+\cdots +a_{1}s+a_{0}=0$.