MHB What is the Intuition Behind Integral Over in Commutative Algebra?

[A.Magnus]

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 Wikipedia 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 $n ≥ 1$ 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.

 

[Opalg]

The motivating example comes from the ring of real numbers. You probably know that every real number is either algebraic (like $\sqrt2$) or transcendental (like $\pi$). The algebraic numbers are those that satisfy a polynomial equation with rational coefficients. So they are exactly the real numbers that are integral over $\Bbb Q$. The set $\Bbb A$ of all algebraic numbers is a ring (in fact, a field – see here), and the ring $\Bbb A$ is an integral extension of $\Bbb Q$.

 

[A.Magnus]

The motivating example comes from the ring of real numbers. You probably know that every real number is either algebraic (like $\sqrt2$) or transcendental (like $\pi$). The algebraic numbers are those that satisfy a polynomial equation with rational coefficients. So they are exactly the real numbers that are integral over $\Bbb Q$. The set $\Bbb A$ of all algebraic numbers is a ring (in fact, a field – see here), and the ring $\Bbb A$ is an integral extension of $\Bbb Q$.

Thank you for your gracious helping hand, apologize for getting back to you late. ~MA