Polynomial rings over Goldie-Kerr commutative rings II

An overlooked corollary to the main result of the stated paper (Proc. Amer. Math. Soc. 120 (1994), 989--993) is that any Goldie ring of Goldie dimension 1 has Artinian classical quotient ring , hence is a Kerr ring in the sense that the polynomial ring satisfies the on annihilators . More generally, we show that a Goldie ring has Artinian when every zero divisor of has essential annihilator (in this case is a local ring; see Theorem ). A corollary to the proof is Theorem 2: A commutative ring has Artinian iff is a Goldie ring in which each element of the Jacobson radical of has essential annihilator. Applying a theorem of Beck we show that any ring that has Noetherian local ring for each associated prime is a Kerr ring and has Kerr polynomial ring (Theorem 5).

     

Exotic differentiable structures and general relativity

We review recent developments in differential topology with special concern for their possible significance to physical theories, especially general relativity. In particular we are concerned here with the discovery of the existence of non-standard (fake or exotic) differentiable structures on topologically simple manifolds such asS ⁷, ⁴ andS ³ X ¹. Because of the technical difficulties involved in the smooth case, we begin with an easily understood toy example looking at the role which the choice of complex structures plays in the formulation of two-dimensional vacuum electrostatics. We then briefly review the mathematical formalisms involved with differentiable structures on topological manifolds, diffeomorphisms and their significance for physics. We summarize the important work of Milnor, Freedman, Donaldson, and others in developing exotic differentiable structures on well known topological manifolds. Finally, we discuss some of the geometric implications of these results and propose some conjectures on possible physical implications of these new manifolds which have never before been considered as physical models.     

Photoelectron spectroscopy of phosphorus hydride anions

Negative-ion photoelectron spectroscopy is applied to the PH-, PH2-, P2H-, P2H2-, and P2H3-molecular anions. Franck-Condon simulations of the photoelectron spectra are used to analyze the spectra and to identify various P2H(n)- species. The simulations employ density-functional theory calculations of molecular geometries and vibrational frequencies and normal modes, and coupled-cluster theory calculations of electron affinities. The following electron affinities are obtained: EA0(PH) = 1.027 +/- 0.006 eV, EA0(PH2) = 1.263 +/- 0.006 eV, and EA0(P2H) = 1.514 +/- 0.010 eV. A band is identified as a mixture of trans-HPPH- and cis-HPPH-. Although the trans and cis bands cannot be definitively assigned from experimental information, using theory as a guide we obtain EA0(trans-HPPH)= 1.00 +/- 0.01 eV and EA0(cis-HPPH) = 1.03 +/- 0.01 eV. A weak feature tentatively assigned to P2H3- has a vertical detachment energy of 1.74 eV. The derived gas-phase acidity of phosphine is delta(acid)G298(PH3) < or = 1509.7 +/- 2.1 kJ mo1(-1).     

Investigation by infrared absorption of the by-products of the cyanoethylation of cotton cellulose

The soluble cyanoethyl ether of cellulose, prepared by reaction in the presence of NaOH catalyst to high degrees of substitution (DS = 2.95), can be precipitated by various organic media and contains more nitrogen than can be accounted for by simple addition of acrylonitrile at each cellulose hydroxyl group. Infrared absorption spectra of such samples and of the reacted residues show various types of vibrational absorption bands, characteristic of amino groups. At advanced stages of the reaction the characteristic broad unresolved band of cellulose extending from 1200–950 cm.^?1 largely disappears, leaving only a few weaker bands which likely overlap those due to C–N stretching. Comparison is made of the spectra of the partially cyanoethylated cellulose and the polyacrylonitrile which forms in the stock acrylonitrile, to characterize the products. It is inferred that the highly cyanoethylated cellulose, partly dissolved in the acrylonitrile, further degrades and changes under a complex mechanism to various forms of amines. The yellow to orange color of the samples is assumed to be due to these by-products. The thermogravimetric analysis reveals that these products are more heat-resistant than the cyanoethylated cellulose.     

Sterols with Cyclopropane-Containing Side Chains: Synthesis and Acid Isomerization

To test the eventual involvement of sterols with a cyclopropane containing side chain in bio-methylation processes, 24,25-methanocholest-5-en-3β-ol ( 5 ) and 24,25-methano-26-methylcholest-5-en-3β-ol ( 6 ) were synthesized. Together with the naturally occurring cyclopropane petrosterol ( 1 ), they were submitted to acid isomerization to provide chemical models for some of their potential metabolites.     

Mass spectrometry in structural and stereochemical problems—CLXXXVII A study of skeletal rearrangements in chromans by combined 13C and deuterium labeling

Chroman (I), 2,2-dimethylchroman (II), flavan (III) and 2,2-diphenylchroman (IV) were labeled with ¹³C or deuterium at every position of importance. The mass spectra of these labeled compounds make possible a detailed interpretation of some very complex carbon skeleton rearrangements. The combination of deuterium and ¹³C-labeling revealed that loss of CH₃, C₂H₅ and C₇H₇ fragments from chroman (I), 2,2-dimethylchroman (II) and 2,2-diphenylchroman (IV) respectively, occurs by multiple pathways involving aryl or alkyl migrations with little hydrogen randomization prior to fragmentation. Although ejection of C₇H₇ from flavan (III) occurs to a small extent by a route similar to those mentioned above, its [M ? C₇H₇] fragment can also be formed by a pathway involving no phenyl migration but extensive hydrogen scrambling. The validity of the schemes proposed on the basis of the ¹³C labeling data was checked by using them to predict the spectra of the deuterium labeled compounds.