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On Commutative FDF-Rings

Mamadou Baïlo Barry, Papa Cheikhou Diop

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Abstract

Let R be a commutative ring with non-zero identity and M be a unital Rmodule. Then M is called Dedekind finite if whenever N is a submodule of M such that M is isomorphic to the module M ⊕ N, then N = 0. The ring R is called FD F-ring if any Dedekind finite R-module is finitely generated. In this note, we show that a commutative ring R is an FD F-ring if and only if it is an Artinian principal ideal ring.

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What this paper is about

Let R be a commutative ring with non-zero identity and M be a unital Rmodule. Then M is called Dedekind finite if whenever N is a submodule of M such that M is isomorphic to the module M ⊕ N, then N = 0. The ring R is called FD F-ring if any Dedekind finite R-module is finitely generated. In this note, we show that a commutative ring R is an FD F-ring if and only if it is an Artinian principal ideal ring.

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Available abstract

Let R be a commutative ring with non-zero identity and M be a unital Rmodule. Then M is called Dedekind finite if whenever N is a submodule of M such that M is isomorphic to the module M ⊕ N, then N = 0. The ring R is called FD F-ring if any Dedekind finite R-module is finitely generated. In this note, we show that a commutative ring R is an FD F-ring if and only if it is an Artinian principal ideal ring.

Key concepts: Principal ideal ring, Commutative ring, Mathematics, Ring (chemistry), Reduced ring, Noncommutative ring, Dedekind cut, Commutative property

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