Locally Dedekind Rings
There exist integral domains which are locally but not globally Dedekind: the localization of at each maximal ideal is a Dedekind ring (equivalently, a DVR) but itself is not Dedekind. As mentioned above, such a ring cannot be Noetherian. It seems that the first examples of such rings were constructed by N. Nakano in 1953. In the literature such rings are sometimes called "proper almost Dedekind rings."
Read more about this topic: Dedekind Domain
Famous quotes containing the words locally and/or rings:
“To see ourselves as others see us can be eye-opening. To see others as sharing a nature with ourselves is the merest decency. But it is from the far more difficult achievement of seeing ourselves amongst others, as a local example of the forms human life has locally taken, a case among cases, a world among worlds, that the largeness of mind, without which objectivity is self- congratulation and tolerance a sham, comes.”
—Clifford Geertz (b. 1926)
“We will have rings and things, and fine array,
And kiss me, Kate, we will be married o’ Sunday.”
—William Shakespeare (1564–1616)