PSEUDO-C * -INJECTIVE AND CO-HOPFIAN MODULES

: The pseudo- * c -injective modules and rings have been introduced in [7]. In this paper, we study co-Hopfian pseudo- * c -injective modules. The main result of this paper is to give sufficient conditions which pseudo- * c -injective modules will be co-Hopfian. We show that the following conditions are equivalent for a pseudo- * c -injective modue M :


Introduction
Throughout the paper, R represents an associative ring with identity 10  and all modules are unitary right R -modules. We write R M (or shoter, M ) to indicate that M is a right R -module. If N is a submodule of M (resp., proper submodule) we denote by NM  (resp., < NM   [3] introduced the concept of the Hopfian module. The dual of Hopficity, i.e., the concept of the co-Hopficity was given lately by Varadarajan [8]. A module is called Hopfian (resp., co-Hopfian) if every epimorphic (resp., monomorphic) endomorphism is an automorphism. The Hopfian (resp., co-Hopfian) concept is an extension of the Noetherian (resp., Artinian) concept. These modules have been investigated by many authors, e.g., W. M. Xue [9], G. Yang and Z. K. Liu [10], D. A. Kamran and M. Amir [5]... A module M is said to be weakly co-Hopfian if every injective R -endomorphism : f M M  is essential, i.e., () e f M M  [10]. In [10], G. Yang and Z. K. Liu proved that if R R is co-Hopfian (or weakly co-Hopfian) then R R is Hopfian.
According to [6] In this paper, we proved that a pseudo-* c -injective module is co-Hopfian if and only if it is directly finite (Theorem 2.10), a pseudo-* c -injective and weakly co-Hopfian module is co-Hopfian (Proposition 2.6) and a pseudo-* c -injective module has the cancellation property then it is co-Hopfian (Corollary 2.9).

Results
By definition, the class of pseudo-* c -injective modules is an extension of the class of pseudoinjective modules. A module N is said to be pseudo- [2]. Lemma 2.1 and Example 2.2 show that the class of pseudo-* c -injective modules is a proper extension of the class of continuous modules and the class of pseudo-injective modules.

Let
D be a right PCI -domain, that is not a division ring. Assume that () ED is the injective hull of D . Then ( ) / E D D is semisimple, () ED has a maximal submodule M containing D . Therefore M is a continuous right D -module but it is not pseudo-injective (see [2]. Thus M is pseudo-* c -injective but it is not pseudo-injective.

Let
M be a module whose lattice of submodules is In which, the single arrows are only proved for the module R R .

1) Let the ring
Then R is co-Hopfian as Rmodule and not generalized Artinian.
2) The -module = pp M   is both generalized Noetherian and generalized Artinian, but it is neither Noetherian nor Artinian, where is the set of all primes.
The following proposition show that all pseudo-* c -injective weakly co-Hopfian modules are co-Hopfian.  M is co-Hopfian.
The following results are some sufficient conditions for a pseudo-* c -injective module will be co-Hopfian.   M is a pseudo-M -* c -injective module and non-co-Hopfian.