• Title/Summary/Keyword: semisimple-M-injective

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On Injectivity of Modules via Semisimplicity

  • Nguyen, Thi Thu Ha
    • Kyungpook Mathematical Journal
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    • v.62 no.4
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    • pp.641-655
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    • 2022
  • A right R-module N is called pseudo semisimple-M-injective if for any monomorphism from every semisimple submodule of M to N, can be extended to a homomorphism from M to N. In this paper, we study some properties of pseudo semisimple-injective modules. Moreover, some results of pseudo semisimple-injective modules over formal triangular matrix rings are obtained.

INJECTIVE PROPERTY RELATIVE TO NONSINGULAR EXACT SEQUENCES

  • Arabi-Kakavand, Marzieh;Asgari, Shadi;Tolooei, Yaser
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.559-571
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    • 2017
  • We investigate modules M having the injective property relative to nonsingular modules. Such modules are called "$\mathcal{N}$-injective modules". It is shown that M is an $\mathcal{N}$-injective R-module if and only if the annihilator of $Z_2(R_R)$ in M is equal to the annihilator of $Z_2(R_R)$ in E(M). Every $\mathcal{N}$-injective R-module is injective precisely when R is a right nonsingular ring. We prove that the endomorphism ring of an $\mathcal{N}$-injective module has a von Neumann regular factor ring. Every (finitely generated, cyclic, free) R-module is $\mathcal{N}$-injective, if and only if $R^{(\mathbb{N})}$ is $\mathcal{N}$-injective, if and only if R is right t-semisimple. The $\mathcal{N}$-injective property is characterized for right extending rings, semilocal rings and rings of finite reduced rank. Using the $\mathcal{N}$-injective property, we determine the rings whose all nonsingular cyclic modules are injective.

Rings Whose Simple Singular Modules are PS-Injective

  • Xiang, Yueming;Ouyang, Lunqun
    • Kyungpook Mathematical Journal
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    • v.54 no.3
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    • pp.471-476
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    • 2014
  • Let R be a ring. A right R-module M is PS-injective if every R-homomorphism $f:aR{\rightarrow}M$ for every principally small right ideal aR can be extended to $R{\rightarrow}M$. We investigate, in this paper, rings whose simple singular modules are PS-injective. New characterizations of semiprimitive rings and semisimple Artinian rings are given.