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2019•Asian-European Journal of MathematicsRequires access

Characterization of rings using finite-direct-injective modules

Sanjeev Kumar Maurya, Ashok Ji Gupta

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Abstract

In this paper, we characterize strongly right [Formula: see text]-rings in terms of finite-direct-injective modules which is a generalization of direct-injective modules (or [Formula: see text]-modules). Using this result, we give an example of a finite-direct-injective module which is not a direct-injective module. We prove that if every finite-direct-injective right [Formula: see text]-module is a direct-injective module, then the ring [Formula: see text] must be right Noetherian. Also, we characterize semisimple artinian rings, regular right FGC-rings in terms of finite-direct-injective modules.

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What this paper is about

In this paper, we characterize strongly right [Formula: see text]-rings in terms of finite-direct-injective modules which is a generalization of direct-injective modules (or [Formula: see text]-modules). Using this result, we give an example of a finite-direct-injective module which is not a direct-injective module. We prove that if every finite-direct-injective right [Formula: see text]-module is a direct-injective module, then the ring [Formula: see text] must be right Noetherian. Also, we characterize semisimple artinian rings, regular right FGC-rings in terms of finite-direct-injective modules.

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Available abstract

In this paper, we characterize strongly right [Formula: see text]-rings in terms of finite-direct-injective modules which is a generalization of direct-injective modules (or [Formula: see text]-modules). Using this result, we give an example of a finite-direct-injective module which is not a direct-injective module. We prove that if every finite-direct-injective right [Formula: see text]-module is a direct-injective module, then the ring [Formula: see text] must be right Noetherian. Also, we characterize semisimple artinian rings, regular right FGC-rings in terms of finite-direct-injective modules.

Key concepts: Injective module, Injective function, Mathematics, Module, Semisimple module, Projective module, Noetherian, Pure mathematics

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