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For a multiplication $R$-module $M$ we consider the Zariski topology in the set $Spec\left( M\right) $ 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\left( M\right) $. 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\left( M/N\right) $ 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 $\mathcal{\uparrow } (N)^{Semp\left( M\right) }=\left\{ P\in Semp\left( M\right) \mid N\subseteq P\right\} $ and the lattice $Semp\left( M/N\right) $ are isomorphic as frames. Finally, we obtain results about quantales and the classical Krull dimension of $M$.
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참고문헌 (31)
참고문헌 신청
Jawad Abuhlail / 2016 / On topological lattices and their applications to module theory / Journal of Algebra and Its Applications 15(03):1650046
T. Albu / 1974 / Sur la dimension de Gabriel des modules / Verlag Uni-Druck
D. D. Anderson / 1980 / Some remarks on multiplication ideals / Math. Japon 25(4):463~469
D.D. Anderson / 2000 / Some remarks on multiplication ideals, ii / Communications in Algebra 28(5):2577~2583
M. F. Atiyah / 1969 / Introduction to Commutative Algebra / Addison-Wesley Publishing Co
T. Albu / 1974 / Sur la dimension de Gabriel des modules / Verlag Uni-Druck
D. D. Anderson / 1980 / Some remarks on multiplication ideals / Math. Japon 25(4):463~469
D.D. Anderson / 2000 / Some remarks on multiplication ideals, ii / Communications in Algebra 28(5):2577~2583
M. F. Atiyah / 1969 / Introduction to Commutative Algebra / Addison-Wesley Publishing Co
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