A comparison of length definitions for maps of modules over local rings

LetM be a finitely generated free module over a local ring. An automorphism ofM can be written as a product of automorphisms that are elements of a given generating groupG of the set of all automorphisms ofM. The minimal number of elements required for such a product is called the length of. We study two decompositions using similar generating groups and compare the two resulting lengths.     

On associative algebras satisfying the Engel condition

It is shown that every finitely generated associative algebra over a field of characteristicp>0 satisfying the Engel condition is Lie-nilpotent. It follows that the Engel condition is inherited from an algebraA to its group of units,U(A).     

On finitely generated multiplication modules

We shall prove that if M is a finitely generated multiplication module and Ann(M) is a finitely generated ideal of R, then there exists a distributive lattice M such that Spec(M) with Zariski topology is homeomorphic to Spec(M) to Stone topology. Finally we shall give a characterization of finitely generated multiplication R-modules M such that Ann(M) is a finitely generated ideal of R.     

Abelian groups flat as modules over their endomorphism ring

Let R be a commutative ring with identity and let M be an R-module. We examine the situation where for each prime ideal ρof R the set of all ρ-prime submodules of M is finite. In case R is Noetherian and M is finitely generated, we prove that this condition is equivalent to there being a positive integer n such that for every prime ideal ρ of R, the number of ρ-prime submodules of Mis less than or equal to n. We further show that in this case, there is at most one ρ-prime submodule for all but finitely many prime ideals ρ of R.     

Nilpotence and generation in the stable module category

Nilpotence has been studied in stable homotopy theory and algebraic geometry. We study the corresponding notion in modular representation theory of finite groups, and apply the discussion to the study of ghosts, and generation of the stable module category. In particular, we show that for a finitely generated kG-module M, the tensor M-generation number and the tensor M-ghost number are both equal to the degree of tensor nilpotence of a certain map associated with M.     

Some Theorems on Leibniz Algebras

If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L and M is a finite-dimensional irreducible L-bimodule, then all U-bimodule composition factors of M are isomorphic. If U is a subnormal subalgebra of a finite-dimensional Leibniz algebra L, then the nilpotent residual of U is an ideal of L. Engel subalgebras of finite-dimensional Leibniz algebras are shown to have similar properties to those of Lie algebras. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements.     

A characterisation of virtually free groups

We prove that a finitely generated group G is virtually free if and only if there exists a generating set for G and k > 0 such that all k-locally geodesic words with respect to that generating set are geodesic. Received: 30 October 2006     

On finiteness of multiplication modules

Our main aim in this note, is a further generalization of a result due to D. D. Anderson, i.e., it is shown that if R is a commutative ring, and M a multiplication R-module, such that every prime ideal minimal over Ann (M) is finitely generated, then M contains only a finite number of minimal prime submodules. This immediately yields that if P is a projective ideal of R, such that every prime ideal minimal over Ann (P) is finitely generated, then P is finitely generated. Furthermore, it is established that if M is a multiplication R-module in which every minimal prime submodule is finitely generated, then R contains only a finite number of prime ideals minimal over Ann (M).      

Central extensions of groups of sections

If K is a Lie group and q : P → M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra \mathfrakk{\mathfrak{k}} of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact 1-forms. In this article, we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components, we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by the specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context, we provide sufficient conditions for integrability in terms of data related only to the group K.     

On finitely generated projective modules

It is shown that the set End _A (M) is a Waelbroeck algebra, for every topological Waelbroeck algebraA and for every finitely generated projectiveA-moduleM. The research of Mart Abel was supported by Greek State Scholarship Foundation and Estonian Science Foundation grant 6205.     

Forcing Linearity Numbers for Multiplication Modules

In this article, we prove that for any multiplication module M, the forcing linearity number of M, fln(M), belongs to {0,1,2}, and if M is finitely generated whose annihilator is contained in only finitely many maximal ideals, then fln(M) = 0. Also, the forcing linearity numbers of multiplication modules over some special rings are given. We also show that every multiplication module is semi-endomorphal.     

Finitely Presented Modules Over Semihereditary Rings

We prove that each almost local-global semihereditary ring R has the stacked bases property and is almost Bézout. More precisely, if M is a finitely presented module, its torsion part tM is a direct sum of cyclic modules where the family of annihilators is an ascending chain of invertible ideals. These ideals are invariants of M. Moreover, M/tM is a projective module which is isomorphic to a direct sum of finitely generated ideals. These ideals allow us to define a finitely generated ideal whose isomorphism class is an invariant of M. The idempotents and the positive integers defined by the rank of M/tM are invariants of M too. It follows that each semihereditary ring of Krull-dimension one or of finite character, in particular each hereditary ring, has the stacked base property. These results were already proved for Prüfer domains by Brewer, Katz, Klinger, Levy, and Ullery. It is also shown that every semihereditary Bézout ring of countable character is an elementary divisor ring.     

Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds

We study Lie group structures on groups of the form C ^∞(M, K), where M is a non-compact smooth manifold and K is a, possibly infinite-dimensional, Lie group. First we prove that there is at most one Lie group structure with Lie algebra for which the evaluation map is smooth. We then prove the existence of such a structure if the universal cover of K is diffeomorphic to a locally convex space and if the image of the left logarithmic derivative in is a smooth submanifold, the latter being the case in particular if M is one-dimensional. We also obtain analogs of these results for the group of holomorphic maps on a complex manifold with values in a complex Lie group K. We further show that there exists a natural Lie group structure on if K is Banach and M is a non-compact complex curve with finitely generated fundamental group.      

Cayley graphs having nice enumerations

LetW be the Cayley graph of an infinite finitely generated group andM be a finite cover ofW. It is proved in the paper thatTh(M) is finitely axiomatizable overW ifW has a nice enumeration (in the sense of G. Ahlbrandt and M. Ziegler). A finitely generated free abelian group provides such an example. It is shown that in the non-abelian case the corresponding examples are rather rate. In particular, in the soluble case they must be virtually abelian. We discuss the finite model property for finite covers of Cayley graphs of virtually abelian groups and the existence of nice enumerations for strongly minimal structures in general.     

QUASI-FROBENIUS BIMODULES OF FUNCTIONS ON A SEMIGROUP

Let _AM_B be a QF-bimodule, A a left Artinian ring, B a right Artinian ring, G a semigroup with a unit element (a monoid). Let M^G be the set of all functions on G with values in M. Consider M^G as an (AG, BG)-bimodule over the semigroup rings AG and BG. It is proved that the annihilator maps I → r_M^G (I) and R → l_AG (R) are mutually inverse bijective Galois correspondences between the set of finitely cogenerated left ideals I ? AG and the set of right BG-submodules R ? M^G finitely generated over B. The maps J → l_M^G (J) and L → r_AG (L) are mutually inverse bijective Galois correspondences between the set of finitely cogenerated right ideals J ? AG and the set of left AG-submodules L ? M^G finitely generated over A. This result also makes it possible, starting from a given QF-bimodule _A M_B , to construct new QF-bimodules _AG/IS_BG/J as bimodules of functions on a semigroup with values in M.     

SUBGROUP DISTORTIONS IN NILPOTENT GROUPS

The explicit formula for the distortion function of a connected Lie subgroup in a connected simply connected nilpotent Lie group is obtained. In particular, we prove that a function f: N → R can be realized (up to equivalence) as the distortion function of a connected Lie subgroup in a connected simply connected nilpotent Lie group if and only if f ～ n^r for some nonnegative r ∈ Q. Considering lattices in Lie groups, we establish the analogous results for finitely generated nilpotent groups.     

An introduction to presentations of monoid acts: quotients and subacts

The purpose of this article is to introduce the theory of presentations of monoids acts. We aim to construct “nice” general presentations for various act constructions pertaining to subacts and Rees quotients. More precisely, given an M-act A and a subact B of A, on the one hand, we construct presentations for B and the Rees quotient A/B using a presentation for A, and on the other hand, we derive a presentation for A from presentations for B and A/B. We also construct a general presentation for the union of two subacts. From our general presentations, we deduce a number of finite presentability results. Finally, we consider the case where a subact B has a finite complement in an M-act A. We show that if M is a finitely generated monoid and B is finitely presented, then A is finitely presented. We also show that if M belongs to a wide class of monoids, including all finitely presented monoids, then the converse also holds.     

Subgroups of prescribed finite index in linear groups

LetR be a commutative ring,M a finitely generatedR-module andG a subgroup of Aut _R M. Under either of the following conditions, for every positive integerd there is a normal subgroupH ofG of finite index such thatG/H contains an element of orderd. (a)G is infinite and finitely generated. (b)R is finitely generated as a ring andG is not unipotent-by-finite. This extends recent work of A. Lubotzky.     

A Regularity Result of Nikolskii Type for an Evolution Equation on Compact Homogeneous Spaces

In the previous paper [2], we proposed a definition of a class of function spaces of Nikolskii type on a special class of riemannian manifolds, namely the manifolds M admitting a transitive compact connected Lie group of isometries. In this paper we continue the subject and we complete the proof that the spaces we have introduced coincide with the classical ones, defined through (finitely many) local charts of M. Moreover, our equivalent definition of these spaces allows us to prove here a regularity result for the solution of a Stefan-like problem on M, in which the initial datum is assumed to be of Nikolskii type.     

Uniform independence in linear groups

We show that for any non-virtually solvable finitely generated group of matrices over any field, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of at most m generators, such that a and b are free generators of a free subgroup. This uniformity result improves the original statement of the Tits alternative.