• East Asian mathematical journal
• /
• v.19 no.2
• /
• pp.251-259
• /
• 2003

The concepts of t-rational extensions and t-essential extensions of modules, where t a preradical for R-Mod, are introduced. The structures of such extensions are determined. Relations between maximal t-rational extensions and other concepts of modules are studied.

• Kyungpook Mathematical Journal
• /
• v.61 no.2
• /
• pp.239-248
• /
• 2021

In this paper we study modules with the W F I⁺-extending property. We prove that if M satisfies the W F I⁺-extending, pseudo duo properties and M/(Soc M) has finite uniform dimension then M decompose into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the W F I⁺-extending, pseudo duo properties and ascending chain (respectively, descending chain) condition on essential submodules then M = M₁ ⊕ M₂ for some semisimple submodule M₁ and Noetherian (respectively, Artinian) submodule M₂. Moreover, we show that if M is a W F I-extending module with pseudo duo, C₂ and essential socle then the quotient ring of its endomorphism ring with Jacobson radical is a (von Neumann) regular ring. We provide several examples which illustrate our results.

• Honam Mathematical Journal
• /
• v.46 no.4
• /
• pp.650-656
• /
• 2024

In this work ⊕ - e-supplemented modules are defined and some properties of these modules are investigated. It is proved that the finite direct sum of ⊕-e-supplemented modules is also ⊕-e-supplemented. Let M be a distributive and ⊕-e-supplemented R-module. Then every factor module and homomorphic image of M are ⊕ - e-supplemented. Let M be a ⊕ - e-supplemented R-module with SSP property. Then every direct summand of M is ⊕ - e-supplemented.

• Kyungpook Mathematical Journal
• /
• v.59 no.3
• /
• pp.391-401
• /
• 2019

A module is said to be ${\pi}$-extending provided that every projection invariant submodule is essential in a direct summand of the module. In this article, we focus on the class of modules having the ${\pi}$-extending property by looking at the singularity of quotient submodules. By doing so, we provide counterexamples, using hypersurfaces in projective spaces over complex numbers, to show that being generalized ${\pi}$-extending is not inherited by direct summands. Moreover, it is shown that the direct sums of generalized ${\pi}$-extending modules are generalized ${\pi}$-extending.

• Kyungpook Mathematical Journal
• /
• v.56 no.4
• /
• pp.1069-1083
• /
• 2016

In this paper, the structure of e-local modules and classes of modules via essentially small are investigated. We show that the following conditions are equivalent for a module M: (1) M is e-local; (2) $Rad_e(M)$ is a maximal submodule of M and every proper essential submodule of M is contained in a maximal submodule; (3) M has a unique essential maximal submodule and every proper essential submodule of M is contained in a maximal submodule.

• Bulletin of the Korean Mathematical Society
• /
• v.42 no.1
• /
• pp.149-155
• /
• 2005

The notion of U-exact sequence (or quasi-exact sequence) of modules was introduced by Davvaz and Parnian-Garamaleky as a generalization of exact sequences. In this paper, we prove further results about quasi-exact sequences. In particular, we give a generalization of Schanuel's Lemma. Also we obtain some relation-ship between quasi-exact sequences and superfluous (or essential) submodules.

• Communications of the Korean Mathematical Society
• /
• v.9 no.2
• /
• pp.271-274
• /
• 1994

The concept of a continuous module is a generalization of that of an injective module, and conditions ($C_1$), (C$_2$) and ($C_3$) are given for this concept in [4]. In this paper, we study modules with properties that are dual to continuity. These will be called discrete and we discuss discrete abelian groups. Throughout R is a ring with identity, M is a module over R, G is an abelian group of finite rank, E is the ring of endomorphisms of G and S is the center of E. Dual to the notion of essential submodules, we define small submodules of a module M over R.(omitted)

• Bulletin of the Korean Mathematical Society
• /
• v.59 no.2
• /
• pp.453-468
• /
• 2022

In this article, we define a module M to be G^z-extending if and only if for each z-closed submodule X of M there exists a direct summand D of M such that X ∩ D is essential in both X and D. We investigate structural properties of G^z-extending modules and locate the implications between the other extending properties. We deal with decomposition theory as well as ring and module extensions for G^z-extending modules. We obtain that if a ring is right G^z-extending, then so is its essential overring. Also it is shown that the G^z-extending property is inherited by its rational hull. Furthermore it is provided some applications including matrix rings over a right G^z-extending ring.