| Affiliation: | Department of Mathematics, Az-Zahra University, Vanak, Post Code 19834, Tehran, Iran -- and -- Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran ; Department of Mathematics, Shahid Beheshti University, Tehran, Iran -- and -- Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran Massoud Tousi ; Department of Mathematics, Shahid Beheshti University, Tehran, Iran -- and -- Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran |
| Abstract: | Let  be a commutative ring with identity and  an  -module. It is shown that if  is pure injective, then  is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if  is Noetherian, then  is pure injective if and only if  is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that  is pure injective if and only if there is a family  of  -algebras which are finitely presented as  -modules, such that  is isomorphic to a direct summand of a module of the form  , where for each  ,  is an injective  -module. |
