We present a simple constructive proof of the fact that every abelian discrete group is uniformly amenable. We improve the growth function obtained earlier and find the optimal growth function in a particular case. We also compute a growth function for some non-abelian uniformly amenable group.
Glaz and Wickless introduced the class of mixed abelian groups which have finite torsion-free rank and satisfy the following three properties: i) is finite for all primes , ii) is isomorphic to a pure subgroup of , and iii) is torsion. A ring is a left Kasch ring if every proper right ideal of has a non-zero left annihilator. We characterize the elements of such that is a left Kasch ring, and discuss related results.
Suppose is a -mixed splitting abelian group and is a commutative unitary ring of zero characteristic such that the prime number satisfies . Then and are canonically isomorphic -group algebras for any group precisely when and are isomorphic groups. This statement strengthens results due to W. May published in J. Algebra (1976) and to W. Ullery published in Commun. Algebra (1986), Rocky Mt. J. Math. (1992) and Comment. Math. Univ. Carol. (1995).
If is an isotype knice subgroup of a global Warfield group , we introduce the notion of a -subgroup to obtain various necessary and sufficient conditions on the quotient group in order for itself to be a global Warfield group. Our main theorem is that is a global Warfield group if and only if possesses an -family of almost strongly separable -subgroups. By an -family we mean an Axiom 3 family in the strong sense of P. Hill. As a corollary to the main theorem, we are able to characterize...
In this paper, we initiate the study of various classes of isotype subgroups of global mixed groups. Our goal is to advance the theory of -isotype subgroups to a level comparable to its status in the simpler contexts of torsion-free and -local mixed groups. Given the history of those theories, one anticipates that definitive results are to be found only when attention is restricted to global -groups, the prototype being global groups with decomposition bases. A large portion of this paper is...
It is well-known that every bounded Abelian group is a direct sum of finite cyclic subgroups. We characterize those non-trivial bounded subgroups of an infinite Abelian group , for which there is an infinite subgroup of containing such that has a special decomposition into a direct sum which takes into account the properties of , and which induces a natural decomposition of into a direct sum of finite subgroups.
Let and be two abelian groups. The group is called -small if the covariant functor commutes with all direct sums and is self-small provided it is -small. The paper characterizes self-small products applying developed closure properties of the classes of relatively small groups. As a consequence, self-small products of finitely generated abelian groups are described.