On co-Hopfian nilpotent groups

A Non-hopfian Group by Gilbert Baumslag

The purpose of this note is to construct a non-hopfian and finitely generated group 5 which is in no way complicated. This group S is a generalised free square of the free nilpotent group A of class two on two generators. Since A satisfies the maximum condition, S differs radically from the non-hopfian groups constructed by Graham Higman [ l ] , with which it may be compared. I t may perhaps be...

On Weakly Co-Hopfian Modules

Generalizations of Hopfian and co-Hopfian modules

In this paper, all rings are associative with identity and all modules are unital left modules unless otherwise specified. Let R be a ring and M a module. N ≤M will mean N is a submodule of M. A submodule E of M is called essential in M (notation E ≤e M) if E∩A = 0 for any nonzero submodule A of M. Dually, a submodule S of M is called small in M (notation S M) if M = S+T for any proper submodul...

Braid Groups Are Almost Co-hopfian

We prove that the braid group on 4 or more strands modulo its center is co-Hopfian. We then show that any injective endomorphism of these braid groups is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from the braid group on n strands to the braid group on n + 1 strands is geometric (n ≥ 7). Additionally, we obtain related r...

Semi Hopfian and Semi Co-Hopfian Modules over Generalized Power Series Rings

Let (S,≤) be a strictly totally ordered monoid, R be a commutative ring and M be an R-module. We show the following results: (1) If (S,≤) satisfies the condition that 0 ≤ s for all s ∈ S, then the module [[MS,≤]] of generalized power series is a semi Hopfian [[RS,≤]]-module if and only if M is a semi Hopfian R-module; (2) If (S,≤) is artinian, then the generalized inverse polynomial module [MS,...