Rings characterized by the extending property for finitely generated submodules

Dung, Banh Duc (2024) Rings characterized by the extending property for finitely generated submodules. Miskolc Mathematical Notes, 25 (2). pp. 659-672. ISSN 1787-2413

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Abstract

A module M is called ef-extending if every closed submodule which contains essentially a finitely generated submodule is a direct summand of M. In this paper, we prove some properties of rings via ef-extending modules and essentially finite injective modules. It is shown that a module M is an ef-extending module and whenever M = H ⊕ K with H essentially finite, then H is essentially finite K-injective if and only if for essentially finite submodules N1,N2 of M with N1 ∩ N2 = 0, there exist submodules M1,M2 of M such that Ni is essential in Mi (i = 1,2) and M1 ⊕ M2 is a direct summand of M. A ring R is right co-Harada if and only if R is right (or left) perfect with ACC on right annihilators and R ⊕ R is ef-extending as a right R-module, iff R is right (or left) perfect and RR(ℕ ⁡ ) is an ef-extending module. Some properties of ef-extending modules over excellent extension rings are considered.

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