Vipul: New page: ==Statement== Suppose $S$ is a commutative unital ring and $R$ is a ''proper'' fact about::integrally closed subring of $S$ . Then, $R$ ...

2009-02-07T02:42:50Z

New page: ==Statement== Suppose <math>S</math> is a commutative unital ring and <math>R</math> is a ''proper'' fact about::integrally closed subring of <math>S</math>. Then, <math>R</math> ...

New page

==Statement==

Suppose <math>S</math> is a [[commutative unital ring]] and <math>R</math> is a ''proper'' [[fact about::integrally closed subring]] of <math>S</math>. Then, <math>R</math> has infinite [[index of a subgroup|index]] in <math>S</math>: in other words, the quotient group <math>S/R</math> is infinite.

==Proof==

'''Given''': A ring <math>S</math>, a proper integrally closed subring <math>R</math> of <math>S</math>.

'''To prove''': <math>R</math> has infinite index in <math>S</math>.

'''Proof''': Suppose <math>R</math> has finite index, say <math>r</math>, in <math>S</math>. Let <math>x</math> be an element in <math>S \setminus R</math>. Consider the elements <math>x,x^2,x^3, \dots, </math>. Since there are only <math>r</math> cosets of <math>R</math> in <math>S</math>, there must exist <math>m > n \ge 1</math> such that <math>x^m, x^n</math> are in the same coset. Let <math>a = x^m - x^n</math>. Then, <math>a \in R</math>, so <math>x</math> satisfies the monic polynomial in <math>R[x]</math>:

<math>x^m - x^n - a = 0</math>.

Proper integrally closed subring has infinite index - Revision history

Vipul: New page: ==Statement== Suppose S is a commutative unital ring and R is a ''proper'' fact about::integrally closed subring of S. Then, R ...

Vipul: New page: ==Statement== Suppose $S$ is a commutative unital ring and $R$ is a ''proper'' fact about::integrally closed subring of $S$ . Then, $R$ ...