Can some one explain what does it mean by the words "integral over $R$"?

Question

The integral closure of a commutative unit ring $R$ in an extension ring $S$ is the set of all elements of $S$ which are integral over $R$. It is a subring of $S$ containing $R$.

Can someone explain what does it mean by the words integral over $R$?

Answer 1

An element $x \in S$ is integral over $R$ if there is a monic polynomial $P(T) = T^n + a_{n-1}T^{n-1}+ \cdots + a_1 T+a_0 \in R[T]$ such that $P(x) = 0.$

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(Non-)examples:

1) Any element of $R$ is integral over $R$.

2) But $1/2 \in S = \Bbb Q$ is not integral over $R = \Bbb Z$ (it is a root of $2T-1$, but this polynomial is not monic ; and no monic polynomial with integer coefficients can have $1/2$ as root).

3) $\pi \in S = \Bbb R$ is not integral over $R = \Bbb Q$.

4) $i \in \Bbb C$ is integral over $\Bbb Z$, being a root of $T^2+1$.

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Geometric interpretation : if $R \subset S$ is integral, then $\mathrm{Spec}(S) \to \mathrm{Spec}(R)$ is closed (because of the going-up), and it is surjective.

Answer 2

An element $s\in S$ is integral over $R$ iff there exists a monic polynomial $f\in R[x]$ such that $f(s) = 0$ (as opposed to agebraic, which doesn't require the polynomial to be monic).

For instance, take $R = \Bbb Z$ and $S = \Bbb R$. Then $\frac12\in \Bbb R$ is not integral, because there is no monic polynomial (with integer coefficients) which has $\frac12$ as root. On the other hand, $\sqrt2$ is integral because $x^2 - 2$ is monic. So the integral closure of $\Bbb Z$ in $\Bbb R$ contains $\sqrt 2$ but not $\frac12$.

Answer 3

the definition of "$S$ is integral over $R$" is well-explained by the two answers already given. however if you are new to this concept it may be helpful to know the significance of this definition.

Informally we may say "$s$ is integral over $R$" means that for any $s \in S$, we have that $R[s]$ is a finite-dimensional extension of $R$. That is to say we can find a finite set $s_1, \dots, s_n$ of elements of $S$ such that any element $s'$ of $R[s]$ can be written as a finite sum: $$ s' = \sum_i r_i s_i $$ with $r_i \in R$.

once you see the connection between this idea and the requirement that $s$ satisfies a monic polynomial over $R$, you will be well on your way to understanding the importance of integrality