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Throughout this paper R is a commutative ring with identity and all modules are unitary. It is well known that an Artinian ring with identity is a Noetherian. But an Artinian module is not Noetherian and a Noetherian module is not Artinian in general. In this paper we have some results for rings whose Artinian modules are Noeteherian modules and Noetherian Modules are Aritinan. Such a rings is called APN-ring.

AMS Subject Classification: 13C13, 13C05