Ring theory MOC

Integral element

Let be a (commutative) ring extension.

An element is called integral over iff there exists a nonzero monic polynomial such that . #m/def/ring We denote the set of all such elements as , and by ^P1 this is a ring.

If every is integral over , then is called integral over , and the extension is called integral.

An integral element over is called an Algebraic integer.

Equivalent conditions

Let . The following are equivalent1

  1. is integral over ;
  2. is a module-finite -ring;
  3. where is some finitely generated extension
Proof

Suppose ^I1 holds with minimal polynomial

Then so is generated by .

If ^I2 is the case, then we get ^I3 trivially with .

Now supposing ^I3 holds, let generate as an -module. Out task is to show that satisfies a monic polynomial . Since , . Letting we have

for some , and therefore

whence by Zero of a matrix over a ring we have for all , implying . Now since where , it follows is the required monic polynomial.

Properties

  1. The set of all integral elements over forms a ring, namely the sum and product of integral elements is again integral.
  2. Integrality is transitive, namely if is integral and is integral, then is integral.
Proof of 1–2

Suppose are integral over . By ^I2 it follows is finitely generated, and since is also integral over it follows is finitely generated. By Finitely generated module over a module-finite -monoid, it follows is finitely generated. Since both

^P1 follows from ^I3.

Suppose is integral and is integral. Then by transitivity of finitely generatedness is integral, proving ^P2.

See also

#state/tidy | #lang/en | #SemBr

Footnotes

  1. 2022. Algebraic number theory course notes, §2.1, p. 9 ↩