Filters and overrings

Abstract

Communicated by G. E. WallLet R be an integral domain. It is well known (see Lambek (1971), Stenstrom (1971)), that idempotent filters of right ideals, torsion radicals and torsion theories are in one-to-one correspondence, but that different idempotent filters !F of right ideals may lead to the same rings of quotiens R?. We have always R cz R? c Qmax(R)-Given this situation one can ask a number of questions. For example: Describe all different idempotent filters for a given ring. Determine all different rings of quotients. When do different filters lead to the same ring of quotients? When are all rings between R and Q maji (R) of the form RgP. When is every R? of the form RS~l, where S is an Ore system? Some of the problems mentioned above are easier to handle if it is possible to use a localizing procedure. This is described in section 1, and one can apply it for example to all noetherian commutative domains (where one knows all different filters). We then consider noncommutative Krull domains. These rings are a generalization of Krull domains, and it is possible to determine all their rings of quotients.It is known (see for example Gilmer and Ohm (1964)) that every ring between a commutative noetherian integral domain R and its field of quotients is always a ring of quotients with respect to some multiplicative system if and only if R is a Dedekind domain with torsion class group. We will give a similar condition for the semigroup of divisorial ideals of a non commutative Krull domain to insure that every ring of quotients with respect to some torsion theory is a ring of quotients with respect to some Ore system.We add some related results about principal ideal domains and Bezont domains in the final section.We will use the word filter instead of idempotent filter and we will use the definition in Stenstrom (1971)

Raman spectroscopic investigation of sodium borosilicate glass structure

“…The frequency shift shows a maximum at x = 0.5 or a plateau at x < 0.5. Many physical properties show a maximum or minimum at M20/B2Os = 1 (where M is a modifier action) [17,18] of borosilicate glass.…”