dedekind ring sentence in Hindi
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- In the literature such rings are sometimes called " proper almost Dedekind rings ."
- A free module is a projective module, but the converse may be wrong over some rings, such as Dedekind rings.
- In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings.
- There exist integral domains R that are locally but not globally Dedekind : the localization of R at each maximal ideal is a Dedekind ring ( equivalently, a DVR ) but R itself is not Dedekind.
- If the ring has special properties, this hierarchy may collapse, e . g ., for any perfect local Dedekind ring, every torsion-free module is flat, projective and free as well.
- The other class of Dedekind rings that is arguably of equal importance comes from geometry : let " C " be a nonsingular geometrically integral "'affine "'algebraic curve over a field " k ".
- For certain common classes of rings, these statements can be reversed ( for example, every torsion-free module over a Dedekind ring is automatically flat and flat modules over perfect rings are always projective ), as is subsumed in the following diagram of module properties:
- Another illustration of the delicate / robust dichotomy is the fact that being a Dedekind domain is, among Noetherian domains, a local domain is a Dedekind ring iff it is a PID iff it is a discrete valuation ring ( DVR ), so the same local characterization cannot hold for PIDs : rather, one may say that the concept of a Dedekind ring is the "'globalization "'of that of a DVR.
- Another illustration of the delicate / robust dichotomy is the fact that being a Dedekind domain is, among Noetherian domains, a local domain is a Dedekind ring iff it is a PID iff it is a discrete valuation ring ( DVR ), so the same local characterization cannot hold for PIDs : rather, one may say that the concept of a Dedekind ring is the "'globalization "'of that of a DVR.
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