# 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.