It is proved that if a Lie ring L admits an automorphism of prime order p with a finite number m of fixed points and with pL = L, then L has a nilpotent subring of index bounded in terms of p and m and whose nilpotency class is bounded in terms of p. It is also shown that if a nilpotent periodic group admits an automorphism of prime order p which has a finite number m of fixed points, then it has a nilpotent subgroup of finite index bounded in terms of m and p and whose class is bounded in terms of p (this gives a positive answer to Hartley's Question 8.81b in the Kourovka Notebook). From this and results of Fong, Hartley, and Meixner, modulo the classification of finite simple groups the following corollary is obtained: a locally finite group in which there is a finite centralizer of an element of prime order is almost nilpotent (with the same bounds on the index and nilpotency class of the subgroup). The proof makes use of the Higman-Kreknin-Kostrikin theorem on the boundedness of the nilpotency class of a Lie ring which admits an automorphism of prime order with a single (trivial) fixed point.

Slender rings and modules.

Lie rings and lie groups admitting an almost regular automorphism of prime order

On reflexive modules and Abelian groups

Slender modules over domains

H-local domains arise in a variety of module- and ideal-theoretic applications. In the first half of the paper, we survey some of these applications, and collect a number of characterizations and examples of h-local domains. In the second half, we show how diverse examples of h-local Pr ¨ ufer domains arise as overrings of Noetherian domains and polynomial rings in finitely many variables.

Characterizations and constructions of h-local domains

For every module M we have a natural monomorphism 
$$\Psi :\coprod\limits_{i \in I} {{\text{Hom}}_R \left( {M,A_i } \right) \to {\text{Hom}}_R \left( {M,\coprod\limits_{i \in I} {A_i } } \right)} $$

 and we focus our attention on the case when Ψ is also an epimorphism. Some other colimits are also considered.

/pdf/modules-commuting-via-hom-with-some-colimits-290vdpcko0.pdf

Modules commuting (via Hom) with some colimits

/pdf/slender-rings-and-modules-3lopen7rxx.pdf

Abelian groups

/pdf/big-projective-modules-are-free-54r4kq564h.pdf

background

Big projective modules are free

/pdf/on-direct-products-of-infinite-cyclic-groups-2sxriv85v3.pdf

On direct products of infinite cyclic groups

/pdf/correction-to-commutants-of-some-hausdorff-matrices-4mrq5ljd2p.pdf

Correction to: ``Commutants of some Hausdorff matrices''.

https://projecteuclid.org/download/pdf_1/euclid.jmsj/1260976880

Slender rings and modules.

Citations

Lie rings and lie groups admitting an almost regular automorphism of prime order

On reflexive modules and Abelian groups

Slender modules over domains

Characterizations and constructions of h-local domains

Modules commuting (via Hom) with some colimits

References

Abelian groups

Big projective modules are free

On direct products of infinite cyclic groups

Correction to: ``Commutants of some Hausdorff matrices''.

Projective modules over weakly noetherian rings

Related Papers (5)

Infinite Abelian groups

Slender modules over domains

Slenderness in abelian categories

Abelian groups

Rings characterized by their cyclic modules