# Whole element

In the mathematical sub-area of commutative algebra , the concept of a whole element in a ring expansion is a generalization of the concept of an algebraic element in a field expansion .

## definition

There was a ring and a - algebra . Then an element is called all over if there is a polynomial with leading coefficient 1, so that applies, i.e. if there are and coefficients with ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle b \ in B}$ ${\ displaystyle A}$ ${\ displaystyle p \ in A [X] \ setminus \ {0 \}}$${\ displaystyle p (b) = 0}$${\ displaystyle n \ in \ mathbb {N}}$${\ displaystyle a_ {0}, a_ {1}, \ dotsc, a_ {n-1} \ in A}$

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

The set of the over whole elements of is called the whole conclusion of in . ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$

If the whole concluding in with matches is completely finished in . However, if the whole ending of in agrees with, so each element is from all over , then is called all over . ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle A}$ ${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle B}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$ ${\ displaystyle A}$

## Examples

• If a ring expansion is , then in particular is an -algebra. If it is all over , one speaks of a whole ring expansion.${\ displaystyle A \ subseteq B}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$
• An integrity ring that is completely closed in its quotient body is called a normal ring .
• The whole closure of the whole numbers in an algebraic number field is called the integral ring of .${\ displaystyle K}$ ${\ displaystyle {\ mathcal {O}} _ {K}}$${\ displaystyle K}$
• If and , then the whole conclusion of in is given as${\ displaystyle A = \ mathbb {Z}}$${\ displaystyle K = \ mathbb {Q} {\ big (} {\ sqrt {5}} {\ big)}}$${\ displaystyle A}$${\ displaystyle K}$
${\ displaystyle {\ mathcal {O}} _ {K} = \ mathbb {Z} \! \ left [{\ frac {1 + {\ sqrt {5}}} {2}} \ right].}$

## Characterization of entire elements in ring extensions

Be an annular extension . Then are equivalent: ${\ displaystyle A \ subseteq B}$${\ displaystyle x \ in B}$

• ${\ displaystyle x}$is all about ,${\ displaystyle A}$
• ${\ displaystyle A [x]}$is finitely generated as a module,${\ displaystyle A}$
• there is a partial ring such that and as a module is finite.${\ displaystyle C \ subseteq B}$${\ displaystyle A [x] \ subseteq C}$${\ displaystyle C}$${\ displaystyle A}$

## properties

• The whole degree of in is a subalgebra of .${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle B}$
• Wholeness is a transitive relation . More precisely, for a ring expansion , it is all over when is all over and all over .${\ displaystyle A \ subseteq B \ subseteq C}$${\ displaystyle C}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle C}$${\ displaystyle B}$
• An algebra is finite if and only if it is finitely generated and whole.${\ displaystyle A}$${\ displaystyle B}$
• Let be a ring expansion, the entire closure of in and a multiplicatively closed subset . Then there is also the whole ending of in , where with denotes the localization after the set .${\ displaystyle A \ subseteq B}$${\ displaystyle C}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle S \ subseteq A}$${\ displaystyle S ^ {- 1} C}$${\ displaystyle S ^ {- 1} A}$${\ displaystyle S ^ {- 1} B}$${\ displaystyle S ^ {- 1}}$${\ displaystyle S}$
• Be a whole ring expansion and zero divisors . Then there is a body if and only if there is a body.${\ displaystyle A \ subseteq B}$${\ displaystyle B}$ ${\ displaystyle A}$${\ displaystyle B}$
• Is a whole ring expansion. Then there is a connection between prime ideal chains in and underlying prime ideal chains in . This is what Cohen-Seidenberg's theorems say .${\ displaystyle A \ subseteq B}$${\ displaystyle B}$${\ displaystyle A}$
• If there is a sub-ring of the body , then the entire closure of is included in the average of all evaluation rings of the .${\ displaystyle A}$${\ displaystyle K}$${\ displaystyle A}$${\ displaystyle K}$${\ displaystyle K}$${\ displaystyle A}$

## literature

• MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Chapter 5, ISBN 0-201-00361-9

## Individual evidence

1. ^ MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Proposition 5.1.
2. ^ MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Corollary 5.4.
3. ^ MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, p. 60
4. ^ MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Proposition 5.6.
5. ^ MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Proposition 5.7.
6. ^ MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Corollary 5.22.