# General Ring Theory/Integral elements and extensions

**Definition (integral element)**:

Let be a ring, and let be a ring extension. An element is called **integral** over if there exist elements such that

- .

**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

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