Torsion points of elliptic curves over large algebraic extensions of finitely generated fields

The following Theorem is proved:Let K be a finitely generated field over its prime field. Then, for almost all e-tuples (σ)=(σ ₁, …,σ _e)of elements of the abstract Galois group G(K)of K we have:

1. If e=1,then E _tor(K(σ))is infinite. Morover, there exist infinitely many primes l such that E(K(σ))contains points of order l.
2. If e≧2,then E _tor(K(σ))is finite.
3. If e≧1,then for every prime l, the group E(K(σ))contains only finitely many points of an l-power order.

HereK(σ) is the fixed field in the algebraic closureK ofK, ofσ ₁, …,σ _e, and “almost all” is meant in the sense of the Haar measure ofG(K).     

Monomorphisms of finitely generated free groups have finitely generated equalizers

Summary The authors' approach to Gersten's graphical model of an automorphism is generalized to prove the theorem of the title, resolving a conjecture of Stallings.     

A pure subalgebra of a finitely generated algebra is finitely generated

We prove the following. Let be a Noetherian commutative ring, a finitely generated -algebra, and a pure -subalgebra of . Then is finitely generated over .

     

Tits geometries and parabolic systems in finitely generated groups II

Mathematische Zeitschrift -     

Factorization in finitely generated domains

Let D be a Noetherian domain. Then it is well known that D is atomic, i.e. every non-zero non-unit a∈D possesses a factorization

     

Fuzzy finitely generated modules

We introduce the category of fuzzy modules and discuss the construction of fuzzy finitely generated modules. Characterizations of a sort of fuzzy modules, in which the fuzzy value distribution is finite, are partly obtained.     

A finitely generated just-infinite profinite branch group which is not positively finitely generated

We construct a finitely generated profinite branch group which is just-infinite and not positively finitely generated.     

Intersections of finitely generated clones

We answer negatively the following question asked by Stanley Burris: Is the intersection of two finitely generated clones always finitely generated?     

Polynomial cycles in finitely generated domains

LetR be a finitely generated domain of zero characteristics, and letN be a given integer. We show that the lengths of all cycles of polynomial mappingsR ^NR ^N defined overR are bounded by a number depending only onR andN.The work of the second author has been supported by the KBN-grant 2-1037-91-01.     

Dominions in finitely generated nilpotent groups

In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of a finitely generated nilpotent group of class two is trivial in the category of all metabelian nilpotent groups.     

Nontrivial identities in finitely generated semigroups

Ivanovo. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 3, pp. 191–197, May–June, 1989.     

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.     

Smooth distributions are finitely generated

A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle.     

Strongly taut finitely generated monoids

A strongly taut monoid is a monoid in which all the powers of any element of the monoid have the same elasticity, that is, the ratio between the maximum and the minimum length of the factorizations of an element remains unchanged under powers. We give a procedure to determine if a finitely generated commutative monoid is strongly taut.