GENERALIZATIONS OF HOPFIAN AND CO-HOPFIAN MODULES

Let R be a ring and M a left R-module. M which satisfies DCC on essential submodules is GCH, and M which satisfies ACC on small submodules is WH. 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.


Introduction and preliminaries
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; 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].
Lemma 1.1 (see [6, 17.3(2)]).Let K, L, and M be modules.Then two monomorphisms f : K → L, g : L → M are essential if and only if g f is essential.

Modules whose essential injective endomorphisms are isomorphic
Let M be a module.M is said to be a generalized co-Hopfian module (GCH module) if any essential injective endomorphism of M is isomorphic.Proposition 2.1.Suppose that M satisfies the condition that N is GCH for every proper essential submodule.Then M itself is GCH.
Proof.Suppose on the contrary that M is not GCH, then there exists an essential injective homorphism g : M → M which is not an isomorphism.Let N = Img.Then N = M and g induces an isomorphism ḡ : M → N. Then ḡ| N : N → N is an essential injective morphism which is not an isomorphism.This is a contradiction since N is a GCH module.
Proposition 2.2.Let M be a GCH module and K be a direct summand of M. Then K is GCH.
and hence f is an isomorphism, as required.
and hence f is surjective, as desired.
It is easy to know that a module M is co-Hopfian if and only if M is both a weakly co-Hopfian module and a GCH module.In [2], Haghany and Vedadi proved that if DCC holds on nonessential submodules of M, then M is weakly co-Hopfian.We also know that Artinian modules are co-Hopfian modules.Thus it is natural that we consider a module with DCC on essential submodules.Theorem 2.4.Let M be a module with DCC on essential submodules.Then M is GCH.
Im f is a descending chain on essential submodules of M by Lemma 1.1.Since M satisfies DCC on essential submodules, there exists n such that Im . Thus f is surjective, as required.

Lemma 2.5. A quasicontinuous module M is continuous if and only if it is a GCH module.
A continuous module is a quasicontinuous module, but a quasicontinuous module need not be a continuous module.Thus the following result gives a sufficient condition such that a quasicontinuous module is a continuous module.
Corollary 2.6.Let M be a module with DCC on essential submodules.Then M is quasicontinuous if and only if M is continuous.

Yongduo Wang 1457
Proof.It follows from Theorem 2.4 and Lemma 2.5.
Proposition 2.7.Let P be a property of modules preserved under isomorphism.If a module M has the property P and satisfies DCC on essential submodules with property P, then M is GCH.
Proof.Suppose that M is not GCH.Then there exists a proper essential submodule N 1 of M with N 1 M. Thus N 1 is not GCH and enjoys P. We have a proper essential submodule N 2 of N 1 with N 2 N 1 .Clearly, N 2 is not GCH and satisfies P. Repeating, we obtain a strictly descending chain N 1 > N 2 > ••• of proper essential submodules each with property P, a contradiction.
Corollary 2.8.If M has DCC on its non-GCH submodules, then M is GCH.
Proof.Suppose not, and let P be the property of being non-GCH.Applying Proposition 2.7, we arrive at a contradiction.Thus M must be GCH.
Example 2.9.A semisimple module M is weakly co-Hopfian if and only if any homogeneous component of M is finitely generated (see [2,Corollary 1.12]).Thus a semisimple module need not be a weakly co-Hopfian module, and hence it is not a co-Hopfian module.However, any semisimple module is GCH.
Let M be a module.The elements of M[X] are formal sums of the form a 0 + a 1 X + ••• + a k X k with k an integer greater than or equal to 0 and a i ∈ M. We denote this sum by Σ k i=1 a i X i (a 0 X 0 is to be understood as the element a 0 ∈ M).Addition is defined by adding the corresponding coefficients.The R[X]-module structure is given by where c µ = Σ i+ j=µ λ i a j , for any λ i ∈ R, a j ∈ M.

Modules whose small surjective endomorphisms are isomorphic
Let M be a module.M is said to be a weakly Hopfian module (WH module) if any small surjection of M is isomorphic.
Theorem 3.1.Let M be a module.If M satisfies the condition that M/N is WH for every small submodule, 0 = N ≤ M. Then M itself is WH.
we will show that Im f [X] ≤ e M[X].It is easy to verify that Im f [X] = (Im f )[X].Since Im f ≤ e M, Im f [X] ≤ e M[X] by Lemma 2.10.Thus f [X] is isomorphic by assumption.The surjectivity of f follows by that of f [X].This completes the proof.