• 제목/요약/키워드: semiprime module

검색결과 12건 처리시간 0.016초

SOME ASPECTS OF ZARISKI TOPOLOGY FOR MULTIPLICATION MODULES AND THEIR ATTACHED FRAMES AND QUANTALES

  • Castro, Jaime;Rios, Jose;Tapia, Gustavo
    • 대한수학회지
    • /
    • 제56권5호
    • /
    • pp.1285-1307
    • /
    • 2019
  • For a multiplication R-module M we consider the Zariski topology in the set Spec (M) of prime submodules of M. We investigate the relationship between the algebraic properties of the submodules of M and the topological properties of some subspaces of Spec (M). We also consider some topological aspects of certain frames. We prove that if R is a commutative ring and M is a multiplication R-module, then the lattice Semp (M/N) of semiprime submodules of M/N is a spatial frame for every submodule N of M. When M is a quasi projective module, we obtain that the interval ${\uparrow}(N)^{Semp}(M)=\{P{\in}Semp(M){\mid}N{\subseteq}P\}$ and the lattice Semp (M/N) are isomorphic as frames. Finally, we obtain results about quantales and the classical Krull dimension of M.

A NOTE ON SIMPLE SINGULAR GP-INJECTIVE MODULES

  • Nam, Sang Bok
    • Korean Journal of Mathematics
    • /
    • 제7권2호
    • /
    • pp.215-218
    • /
    • 1999
  • We investigate characterizations of rings whose simple singular right R-modules are GP-injective. It is proved that if R is a semiprime ring whose simple singular right R-modules are GP-injective, then the center $Z(R)$ of R is a von Neumann regular ring. We consider the condition ($^*$): R satisfies $l(a){\subseteq}r(a)$ for any $a{\in}R$. Also it is shown that if R satisfies ($^*$) and every simple singular right R-module is GP-injective, then R is a reduced weakly regular ring.

  • PDF