Could somebody write me the intuition behind the concept of "Integral Over"? Please do not write me its formal definition, I can easily get it from textbook. What I am also looking for is its motivation behind it. Please give me also examples.

For your convenience, the formal definition according to goes like this:

In commutative algebra, an element $b$ of a commutative ring $B$ is said to be integral over $A$, a subring of $B$, if there are $n â‰¥ 1$ and $a_{j}\ in A$ such that

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

That is to say, $b$ is a root of a monic polynomial over $A$. If every element of $B$ is integral over $A$, then it is said that $B$ is integral over $A$, or equivalently $B$ is an integral extension of $A$.

I understand that plain simple English definition runs the risk of imprecision; I will take it as working definition only to be improved as I progress along. Thank you for your times and gracious helping hand.