The Artin-Rees property and homology

The Artin-Rees property for a finitely generated nilpotent groupG is used to prove that for any finitely generatedG-moduleM, where is the completion ofM with respect to the augmentation ideal ofZ[G]. Applications to topology are given. Partially supported by NSF Grant GP 33960X.     

Extending asymmetric convergence and Cauchy condition using ideals

In this paper we use the notion of ideals to extend the convergence and Cauchy conditions in asymmetric metric spaces. The asymmetry (or rather, absence of symmetry) of these spaces makes the whole treatment different from the metric case and we use a genuinely asymmetric condition called (AMA) to prove many results and show that certain classic results fail in the asymmetric context if the assumption is dropped.     

Rings with minimum condition on principal ideals

Ohne Zusammenfassung     

On primitive ideals

We extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the Irreducibility theorem for associated varieties and Duflo theorem on primitive ideals, to much wider classes of algebras. Our general version of the Irreducibility Theorem says that if A is a positively filtered associative algebra such that gr A is a commutative Poisson algebra with finitely many symplectic leaves, then the associated variety of any primitive ideal in A is the closure of a single connected symplectic leaf. Our general version of the Duflo theorem says that if A is an algebra with a triangular structure, see § 2, then any primitive ideal in A is the annihilator of a simple highest weight module. Applications to symplectic reflection algebras and Cherednik algebras are discussed.     

Ditkin's condition and primary ideals in central Beurling algebras

In this note some questions of ideal theory for the center and more generally theB-fixed subalgebras of a Beurling algebraL ¹ (G) are discussed. Sufficient conditions on are given for these subagebras to satisfy Ditkin's condition, or for primary ideals to be maximal or at least of finite codimension.Research partially supported by the National Science Foundation.     

On Hamilton's ideals

A theorem is proved that is, in a sense to be made precise, the best possible generalization of the theorems of Dirac, Pósa, and Bondy that give successively weaker sufficient conditions for a graph to be Hamiltonian. Some simple corollaries are deduced concerning Hamiltonian paths, n-Hamiltonian graphs, and Hamiltonian bipartite graphs.     

An Engel condition with derivation for left ideals

We generalize a number of results in the literature by proving the following theorem: Let be a semiprime ring, a nonzero derivation of , a nonzero left ideal of , and let . If for some positive integers , and all , the identity holds, then either or else the ideal of generated by and is in the center of . In particular, when is a prime ring, is commutative.

     

A condition for diagonalization modulo arbitrary norm ideals

We give a spectral condition which is sufficient for the simultaneous diagonalization of a commuting tuple of self-adjoint operators modulo a given norm ideal. For diagonalization modulo certain norm ideals this condition is also necessary, while for other norm ideals this condition seems to be close to being necessary. Moreover, this condition is easy to verify in applications.     

Desingularizations and the Uniform Artin-Rees Theorem

It is proved that an excellent ring which is the homomorphicimage of a regular ring of finite Krull dimension and havinga resolution of singularities satisfies the uniform Artin–Reesproperty. Extensions of some results of Lipman, Sathaye andTeissier are also obtained.