Two characterizations of pure injective modules

• Published in: Proceedings of the American Mathematical Society Vol. 134; no. 10; pp. 2817 - 2822
• Main Authors: Divaani-Aazar, Kamran, Esmkhani, Mohammad, Tousi, Massoud
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-10-2006
• Subjects: Algebra; College mathematics; Commutative rings and algebras; Direct products; Exact sciences and technology; Homomorphisms; Mathematical rings; Mathematics; Sciences and techniques of general use; Theoretical physics; finitely embedded modules; Pure injective modules; injective cogenerators; finitely presented modules; Injective cogenerator; Embedding; Injective module; Commutative ring; Module
• ISSN: 0002-9939; 1088-6826
• DOI: 10.1090/S0002-9939-06-08336-5

Let R be a commutative ring with identity and D an R-module. It is shown that if D is pure injective, then D is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if R is Noetherian, then D is pure injective if and only if D is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that D is pure injective if and only if there is a family \{T_\lambda\}_{\lambda\in \Lambda} of R-algebras which are finitely presented as R-modules, such that D is isomorphic to a direct summand of a module of the form \prod_{\lambda\in \Lambda}E_\lambda, where for each \lambda\in \Lambda, E_\lambda is an injective T_\lambda-module.