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
- .
The set of the over whole elements of is called the whole conclusion of in .
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 .
Examples
- If a ring expansion is , then in particular is an -algebra. If it is all over , one speaks of a whole ring expansion.
- 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 .
- If and , then the whole conclusion of in is given as
Characterization of entire elements in ring extensions
Be an annular extension . Then are equivalent:
- is all about ,
- is finitely generated as a module,
- there is a partial ring such that and as a module is finite.
properties
- The whole degree of in is a subalgebra of .
- Wholeness is a transitive relation . More precisely, for a ring expansion , it is all over when is all over and all over .
- An algebra is finite if and only if it is finitely generated and whole.
- 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 .
- 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.
- 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 .
- 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 .
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
- ^ MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Proposition 5.1.
- ^ MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Corollary 5.4.
- ^ MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, p. 60
- ^ MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Proposition 5.6.
- ^ MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Proposition 5.7.
- ^ MF Atiyah and IG MacDonald: Introduction to Commutative Algebra. Addison-Wesley Series in Mathematics, 1969, Corollary 5.22.