GK-dimensions of corners and ideals

IfT is a corner in a ringR (i.e.TRT ?T) then GK (T)=GK(RT ² R), and hence GK(ReR)=GK(eRe) for idempotentse. Examples are given to show that GK(eRe) can be any integerR) even for prime rings. Positive results are obtained if GK(R)≦2, andR is Goldie.     

q-analogs of harmonic oscillators and related rings

A ringH _q which is aq-analog of the universal enveloping algebra of the Heisenberg Lie algebraU(h) is constructed, and its ring theoretic properties are studied. It is shown thatH _q has a factor ringA _q which is a simple domain with properties that are compared to the Weyl algebra. A secondq-analogH _q ofU(h) is constructed, andH _q is shown to be a primitive ring.     

Algebras which are nearly finite dimensional and their identities

Suppose that all the nonzero one-sided or two-sided ideals of an algebra have finite codimension. To what extent must the algebra be p.i. or primitive?     

Polynomials over division rings

LetD be a division ring with a centerC, andD[X ₁, …,X _N] the ring of polynomials inN commutative indeterminates overD. The maximum numberN for which this ring of polynomials is primitive is equal to the maximal transcendence degree overC of the commutative subfields of the matrix ringsM _n(D),n=1, 2, …. The ring of fractions of the Weyl algebras are examples where this numberN is finite. A tool in the proof is a non-commutative version of one of the forms of the “Nullstellensatz”, namely, simpleD[X ₁, …,X _m]-modules are finite-dimensionalD-spaces. This paper was written while the authors were Fellows of the Institute for Advanced Studies, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, Israel.     

Linear structures on the collections of minimal surfaces inℝ 3 andℝ 4

The collection of minimal herissons in ³ is endowed with a vector space structure. The existence of this structure is related to the fact that null curves inC ³ are described by a single map from the étalé space of the sheaf of germs of holomorphic sections of the line bundle of degree 2 over ₁ to C³, which islinear on stalks. There is an analogous construction for null curves inC ⁴. This gives a similar class of minimal surfaces in ⁴.     

Comparative dimerization of 1-butene with a variety of metal catalysts, and the investigation of a new catalyst for C=H bond activation

Catalytic dimerization of 1-butene by a variety of catalysts is carried out, and the products are analyzed by gas chromatography and mass spectrometry. Catalysts based on cobalt and iron can produce highly linear dimers, with the cobalt-based dimers exceeding 97 % linearity. Catalysts based on vanadium and aluminum prefer to make branched dimers, which are most often methyl-heptenes in the case of vanadium and almost exclusively 2-ethyl-1-butene in the case of aluminum. The vanadium catalyst also produces substantial amounts of dienes and alkanes, suggesting a competing hydrogenation/dehydrogenation pathway that appears to involve vinyl Cbond;H bond activation. Nickel catalysts are generally less selective than those based on iron or cobalt for making linear dimers, but they can make dimers with 60 % linearity. The major by-products for the nickel systems are trisubstituted internal olefins. An important side reaction that must be considered for dimerization reactions is 1-butene isomerization to 2-butene, which makes recycling the butene difficult for a linear dimerization process. Aluminum, iron, and vanadium systems promote very little isomerization, but nickel and cobalt systems tend to isomerize the undimerized substrate heavily.