Addendum to “examples of semiperfect rings”

We fill a gap in [4], and provide a rigorous example of a local ringR whose Jacobson radical is locally nilpotent, butM ₂(R) is not strongly π-regular. The online version of the original article can be found at The work of the first author was partially supported by the DGICYT (Spain), through the grant PB92-0586, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.     

Probing the formation of bicyclo[4.2.0]octan-1-ols

Reaction of lithium enolates of simple ketones with (+/-)-phenyl vinyl sulfoxide has potential for the convergent construction of complex fused ring systems containing a bicyclo[n.2.0]alkan-1-ol. The formation of sulfinylbicyclo[4.2.0]octan-1-ols 1-3 from the lithium enolate of cyclohexanone with (+/-)-phenyl vinyl sulfoxide or (R)-(+)-p-tolyl vinyl sulfoxide 18 was used to probe the mode of this novel cyclization reaction. Using phenyl vinyl sulfoxide, variations in the reaction lighting and solvent were investigated, in conjunction with radical trapping (TEMPO) and isotope labeling (deuterium) experiments. Cyclization to form sulfinylbicyclooctanols 1-3 is likely to proceed via an intermediate that ring closes to the bicycloalkanol anion 11 and was presently favored by the use of solvents such as THF or DME.     

Synthesis and structure of FeRu3(CO)13(μ-PPh2)2; A “64-electron” cluster complex with a planar triangulated rhomboidal array of metal atoms

The species FeRu₃(CO)₁₃(μ-PPH₂)₂, synthesized from Ru₃(CO)₁₂ and Fe(CO)₄(Ph₂PPPh₂),has been characterized both spectroscopically and via a single-crystal X-ray structural analysis. This complex crystallizes in the centrosymmetric triclinic space group P$\text{1}$ [No. 2, C_i¹] with a  10.066(3), b  12.899(3), c  17.003(4) Å, α  111.89(2), β  91.02(2), γ  102.00(2)°, V  1992.7(9) ų, Z  2, ?(obsd)  1.79(2) g cm^-3 and ?(calcd)  1.82 cm^-3. Diffraction data were collected with a Syntex P2₁ automated four-circle diffractometer and the structure was refined to R_F  6.0% and R_WF  3.6% for all 5213 reflections (R_F  3.8%, R_WF  3.6% for those 4140 reflections with |F_o|> 3σ(|F_o|).The metal atoms define a planar triangulated rhombus, with atoms Ru(1) and Ru(2) at the bridgehead, and Fe(1) and Ru(3) at the acute apices. Fe(1) is linked to four terminal carbonyl ligands and is associated with the heteronuclear bonds Fe(1)Ru(1)  2.861(1) Å and Fe(1)Ru(2)  2.868(1) Å. The ruthenium atoms are each bonded to three terminal carbonyl groups. The retheniumruthenium distances are Ru(1)Ru(2)  3.098(1), Ru(1)Ru(3)  3.147(1), and Ru(2)Ru(3)  3.171(1) Å. The structure is completed by Ph₂P bridges across the Ru(1)Ru(3) and Ru(2)(ru(3) vectors (<Ru(1)P(1)Ru(3)  84.89(5)° and <Ru(2)P(2)Ru(3)  85.56(6)°).     

Heavy-Atom Tunneling in Semibullvalenes: How Driving Force,Substituents, and Environment Influence the Tunneling Rates

The Cope rearrangement of selectively deuterated isotopomers of 1,5-dimethylsemibullvalene 2 a and 3,7-dicyano-1,5-dimethylsemibullvalene 2 b were studied in cryogenic matrices. In both semibullvalenes the Cope rearrangement is governed by heavy-atom tunneling. The driving force for the rearrangements is the small difference in the zero-point vibrational energies of the isotopomers. To evaluate the effect of the driving force on the tunneling probability in 2 a and 2 b , two different pairs of isotopomers were studied for each of the semibullvalenes. The reaction rates for the rearrangement of 2 b in cryogenic matrices were found to be smaller than the ones of 2 a under similar conditions, whereas differences in the driving force do not influence the rates. Small curvature tunneling (SCT) calculations suggest that the reduced tunneling rate of 2 b compared to that of 2 a results from a change in the shape of the potential energy barrier. The tunneling probability of the semibullvalenes strongly depends on the matrix environment; however, for 2 a in a qualitatively different way than for 2 b .     

Central simple algebras

Wedderburn’s factorization of polynomials over division rings is refined and used to prove that every central division algebra of degree 8, with involution, has a maximal subfield which is a Galois extension of the center (with Galois group Z₂⊕Z₂⊕Z₂). The same proof, for an arbitrary central division algebra of degree 4, gives an explicit construction of a maximal subfield which is a Galois extension of the center, with Galois group Z₂⊕Z₂. Use is made of the generic division algebras, with and without involution. This work was supported by the Israel Committee for Basic Research and the Anshel Pfeffer Chair.     

Cyclic division algebras

A general example of cyclic division algebra is given, based on a construction of Brauer, yielding examples of division algebras of arbitrary prime exponent without proper central subalgebras, and also noncrossed products of arbitrary exponent. This research was supported in part by the U.S.-Israel Binational Science Foundation. An erratum to this article is available at .     

直接键连原子间的NMR偶合常数的自然杂化轨道研究V.~(13)C-~(13)C和~(13)C-~(31)P核自旋偶合常数~1J_(CC)和~1J_(CP)

在前文工作的基础上，结合ＭＮＤＯ／ＥＨＭＯ分子轨道方法和自然杂化轨道方法，具体计算了ＣＣ键和ＣＰ键的核自旋偶合常数．计算结果表明，１ＪＣＣ和１ＪＣＰ主要由成键原子的轨道杂化作用和键极性这两种结构因素所决定．为从简单价键理论角度解释和计算１ＪＣＣ和１ＪＣＰ值提供了简便直观的方法．     

Division algebras of degree 4 and 8 with involution

We develop necessary and sufficient conditions for central simple algebras to have involutions of the first kind, and to be tensor products of quaternion subalgebras. The theory is then applied to give an example of a division algebra of degree 8 with involution (of the first kind), without quaternion subalgebras, answering an old question of Albert; another example is a division algebra of degree 4 with involution (*) has no (*)-invariant quaternion subalgebras. The research of the second author is supported by the Anshel Pfeffer Chair. The third author would like to express his gratitude to Professor J. Tits for many stimulating conversations.