Generalizations of Hopfian and co-Hopfian modules
نویسنده
چکیده
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 submodule T of M. M is said to be Hopfian (co-Hopfian) in case any surjective (injective) homomorphism is automatically an isomorphism. M is called generalized Hopfian (GH) if any of its surjective endomorphisms has a small kernel. M is called weakly co-Hopfian if any injective endomorphism of M is essential. In this paper, we introduce the concepts of GCH modules and WH modules. It is shown that (1) a module M which satisfies DCC on essential submodules is GCH and a module M which satisfies ACC on small submodules is WH; (2) if M[X] is GCH R[X]-module, then M is GCH R-module. Examples show that a GCH module need not be co-Hopfian and a WH module need not be Hopfian. The notions which are not explained here will be found in [4].
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ورودعنوان ژورنال:
- Int. J. Math. Mathematical Sciences
دوره 2005 شماره
صفحات -
تاریخ انتشار 2005