• 제목/요약/키워드: injective hull

검색결과 6건 처리시간 0.014초

ON THE INJECTIVITY OF THE WEAK TOPOS FUZ

  • Kim, Ig Sung
    • Korean Journal of Mathematics
    • /
    • 제17권2호
    • /
    • pp.161-167
    • /
    • 2009
  • Category Fuz of fuzzy sets has a similar function to the Category Set. We study injective, absolute retract, enough injectives, injective hulls and essential extension in the Category Fuz of fuzzy sets.

  • PDF

A NOTE ON MONOFORM MODULES

  • Hajikarimi, Alireza;Naghipour, Ali Reza
    • 대한수학회보
    • /
    • 제56권2호
    • /
    • pp.505-514
    • /
    • 2019
  • Let R be a commutative ring with identity and M be a unitary R-module. A submodule N of M is called a dense submodule if $Hom_R(M/N,\;E_R(M))=0$, where $E_R(M)$ is the injective hull of M. The R-module M is said to be monoform if any nonzero submodule of M is a dense submodule. In this paper, among the other results, it is shown that any kind of the following module is monoform. (1) The prime R-module M such that for any nonzero submodule N of M, $Ann_R(M/N){\neq}Ann_R(M)$. (2) Strongly prime R-module. (3) Faithful multiplication module over an integral domain.

ON π-V-RINGS AND INTERMEDIATE NORMALIZING EXTENSIONS

  • Min, Kang-Joo
    • 충청수학회지
    • /
    • 제15권2호
    • /
    • pp.35-39
    • /
    • 2003
  • In this paper we study a ring over which every left module of finite length has an injective hull of finite length. We consider a ring that is a finite intermediate normalizing extension ring of such a ring. We also consider the subrings of such a ring.

  • PDF

ON 𝑺-CLOSED SUBMODULES

  • Durgun, Yilmaz;Ozdemir, Salahattin
    • 대한수학회지
    • /
    • 제54권4호
    • /
    • pp.1281-1299
    • /
    • 2017
  • A submodule N of a module M is called ${\mathcal{S}}$-closed (in M) if M/N is nonsingular. It is well-known that the class Closed of short exact sequences determined by closed submodules is a proper class in the sense of Buchsbaum. However, the class $\mathcal{S}-Closed$ of short exact sequences determined by $\mathcal{S}$-closed submodules need not be a proper class. In the first part of the paper, we describe the smallest proper class ${\langle}\mathcal{S-Closed}{\rangle}$ containing $\mathcal{S-Closed}$ in terms of $\mathcal{S}$-closed submodules. We show that this class coincides with the proper classes projectively generated by Goldie torsion modules and coprojectively generated by nonsingular modules. Moreover, for a right nonsingular ring R, it coincides with the proper class generated by neat submodules if and only if R is a right SI-ring. In abelian groups, the elements of this class are exactly torsionsplitting. In the second part, coprojective modules of this class which we call ec-flat modules are also investigated. We prove that injective modules are ec-flat if and only if each injective hull of a Goldie torsion module is projective if and only if every Goldie torsion module embeds in a projective module. For a left Noetherian right nonsingular ring R of which the identity element is a sum of orthogonal primitive idempotents, we prove that the class ${\langle}\mathcal{S-Closed}{\rangle}$ coincides with the class of pure-exact sequences of modules if and only if R is a two-sided hereditary, two-sided $\mathcal{CS}$-ring and every singular right module is a direct sum of finitely presented modules.