On super and hereditarily hopfian and co-hopfian Abelian groups

The notions of Hopfian and co-Hopfian groups have been of interest for some time. In this present work we characterize the more restricted classes of hereditarily Hopfian (co-Hopfian) and super Hopfian (co-Hopfian) groups in the case where the groups are Abelian.     

On some generalizations of Hopfian and co-Hopfian Abelian groups

The notions of Hopfian and co-Hopfian groups have been of interest for some time. In this present work we consider generalizations of the classes of hereditarily Hopfian (co-Hopfian) and super Hopfian (co-Hopfian) Abelian groups by requiring only that countable subgroups or images inherit the Hopfian (co-Hopfian) property.     

Duality for Weakly Co-Hopfian and Generalized Hopfian Modules

Abstract

Duality like connections are established for generalized Hopfian modules and weakly co-Hopfian modules. A generalization of ring Hopficity is also introduced and this concept is studied for matrix rings.     

Notes on Generalized Hopfian and Weakly Co-Hopfian Modules

In this article, the generalized Hopficity and weakly co-Hopficity of modules over truncated polynomial and triangular matrix rings are considered. Also, it is obtained partially when the direct product R × S is a generalized Hopfian ring, where R and S are two rings.     

When is an Abelian group isomorphic to its endomorphism group?

In the paper, necessary and sufficient conditions for an Abelian group A to be isomorphic to the endomorphism group End(A) are obtained. The classes of periodic Abelian groups, divisible Abelian groups, nonreduced Abelian groups, and reduced algebraically compact Abelian groups are considered. For certain classes of Abelian groups, the isomorphism problem for a group and its endomorphism group is solved under the assumption that the endomorphism group itself has the corresponding property.     

Polynomial rings over commutative reduced Hopfian local rings

We prove that if R is a commutative, reduced, local ring, then R is Hopfian if and only if the ring R[x] is Hopfian. This answers a question of Varadarajan [16], in the case when R is a reduced local ring. We provide examples of non-Noetherian Hopfian commutative domains by proving that the finite dimensional domains are Hopfian. Also, we derive some general results related to Hopfian rings.     

Transfinite Diameter with Respect to the Neumann Function

In this paper, we study holomorphically isomorphic Abelian groups, i.e., Abelian groups with isomorphic holomorphs. We also study a generalization of the concept of holomorphic isomorphism, namely, almost holomorphic isomorphism, which is deeply connected with normal Abelian subgroups of holomorphs of Abelian groups. Torsion-free Abelian groups that are determined by their holomorphs are highlighted from different classes. In particular, it has been found that any homogeneous separable group can be determined by its holomorph in the class of all Abelian groups.     

The Hopf Property for Subgroups of Hyperbolic Groups

A group is said to be Hopfian if every surjective endomorphism of the group is injective. We show that finitely generated subgroups of torsion-free hyperbolic groups are Hopfian. Our proof generalizes a theorem of Sela (Topology 35 (2) 1999, 301–321).     

On Pierce Problem for Reduced <Emphasis Type="Italic">p</Emphasis>-Groups

One of the problems in the study of Abelian groups and their endomorphism rings is the problem of constructing an appropriate structural theory. The structural theorems obtained for Abelian groups and their endomorphism rings allowto reduce the study of complex objects to simpler ones. A radical is one of the tools possessing this property. In this paper, we describe the Jacobson radical of the endomorphism rings of reduced p-groups from some class, which is a solution to the Pierce problem for this class of groups.