dedekind ring मीनिंग इन हिंदी

उदाहरण वाक्य

1. In the literature such rings are sometimes called " proper almost Dedekind rings ."
2. A free module is a projective module, but the converse may be wrong over some rings, such as Dedekind rings.
3. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings.
4. 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.
5. 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.
6. 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 ".
7. 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:
8. 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.
9. 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.