Hopfian and co-Hopfian S-acts

In this paper, we consider Hopfian (Co-Hopfian) S-acts; S-acts in which every surjective (injective) endomorphism is an automorphism. Also, we consider strongly Hopfian S-acts (respectively strongly co-Hopfian); an S-acts A is said to be strongly Hopfian (respectively strongly co-Hopfian), if for every endomorphism f of A the chain ker respectively stabilizes. We prove that the class of strongly Hopfian (respectively strongly co-Hopfian) S-acts lies properly between the calss of Noetherian (respectively Artinian) and the class of Hopfian (respectively co-Hopfian) S-acts. One of the main purposes of this paper is to prove this fact that for a quasi projective (respectively quasi injective) S-act A, if A is strongly co-Hopfian (respectively strongly Hopfian) then A is strongly Hopfian (respectively strongly co- Hopfian). Among other things, we also discuss about subsets and quotients of strongly Hopfian and strongly co-Hopfian S-acts.

کلیدواژه ها:

S-act ، Hopfian ، co-Hopfian ، strongly Hopfian ، strongly co-Hopfian ، quasi projective ، quasi injective