dedekind domain वाक्य
उदाहरण वाक्य
मोबाइल
- It is also universally catenary as it is a Dedekind domain.
- Some authors add the requirement that a Dedekind domain not be a field.
- In Dedekind domains, the situation is much simpler.
- These are the prototypical examples of Dedekind domains.
- This notion can be used to study the various characterizations of a Dedekind domain.
- Now let M be a finitely generated module over an arbitrary Dedekind domain R.
- For a Dedekind domain this is of course the same as the ideal class group.
- Over an arbitrary Dedekind domain one has
- In fact, every abelian group is isomorphic to the ideal class group of some Dedekind domain.
- On the other hand, the ring of integers in a number field is always a Dedekind domain.
- Multiplication is defined for ideals, and the rings in which they have unique factorization are called Dedekind domains.
- An immediate consequence of the definition is that every principal ideal domain ( PID ) is a Dedekind domain.
- In fact, this is the definition of a Dedekind domain used in Bourbaki's " Commutative algebra ".
- Similarly, an integral domain is a Dedekind domain if and only if every divisible module over it is injective.
- In fact a Dedekind domain is a unique factorization domain ( UFD ) if and only if it is a PID.
- All Dedekind domains of characteristic 0 and all local Noetherian rings of dimension at most 1 are J-2 rings.
- While every number ring is a Dedekind domain, their union, the ring of algebraic integers, is a Pr�fer domain.
- Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields.
- Applying this theorem when " R " is itself a PID gives us a way of building Dedekind domains out of PIDs.
- There is a version of unique prime factorization for the ideals of a Dedekind domain ( a type of ring important in number theory ).
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