Some aspects of Zariski topology for multiplication modules and their attached frames and quantales
J. Korean Math. Soc.
Published online June 4, 2019
Jaime Castro, José Ríos, and Gustavo Tapia
ITESM, Tlalpan, Instituto de Matemáticas UNAM, Universidad Autónoma de Ciudad Juárez
Abstract : Abstract.

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 [N;M] = {P ∈ Semp (M) | N⊆P and the lattice
Semp (M/N) are isomorphic as frames. Finally, as applications we obtain results
about quantales and the classical Krull dimension of M.
Keywords : Multiplication modules; ; Zariski topology; frames, quantales; classical krull dimension
MSC numbers : 16S90; 16D50; 16P50; 16P70
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