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 . ${\displaystyle A\subseteq B}$ ${\displaystyle C}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle S\subseteq A}$ ${\displaystyle S^{-1}C}$ ${\displaystyle S^{-1}A}$ ${\displaystyle S^{-1}B}$ ${\displaystyle S^{-1}}$ ${\displaystyle S}$

• Total seclusion is a local quality .

• 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.