# General Ring Theory/Integral elements and extensions

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