Geometric Ring Research Articles

Strained cycloalkenynes

Presently known strained cycloalkynes with one, two or three additional cis- or trans-configurated double bonds are summarized in Table 3. The main topics of this article are the geometrical ring strain, the preparation or in situ generation of these compounds by fragmentation of the corresponding 1,2,3-selenadiazoles, and the thermal isomerization processes performed at room temperature or in flash pyrolysis experiments at 440–640°.

A Note on Buchsbaum Rings and Localizations of Graded Domains

Let R = ⊕i ≧0Ri be a graded integral domain, and let p ∈ Proj (R) be a homogeneous, relevant prime ideal. Let R(p) = {r/t| r ∈ Ri, t ∈ Ri\p} be the geometric local ring at p and let Rp = {r/t| r ∈ R, t ∈ R\p} be the arithmetic local ring at p. Under the mild restriction that there exists an element r1 ∈ R1\p, W. E. Kuan [2], Theorem 2, showed that r1 is transcendental over R(p) andwhere S is the multiplicative system R\p. It is also demonstrated in [2] that R(p) is normal (regular) if and only if Rp is normal (regular). By looking more closely at the relationship between R(p) and R(p), we extend this result to Cohen-Macaulay (abbreviated C M.) and Gorenstein rings.

Differential criteria for flatness

We introduce differential algebra methods to the study of flatness over Noetherian domains. The results concern the specific cases of ideals and attempt to use the underlying divisibility properties of the ring. They concern mostly regular (geometric) rings and one-dimensional rings.

Some properties of analytically irreducible geometric quotient rings

The purpose of this note is to show how a considerable part of the local theory of the prime divisors of a field of algebraic functions can be extended to analytically irreducible geometric quotient rings. In doing this we shall make frequent use of Chevalley's paper on local and semi-local rings (2). For brevity this paper will be referred to as ‘L.R.’. We begin the discussion by recalling for the reader's convenience the definition of the Kronecker product of two fields over a common subfield, since this concept will play an important role later.