Communications of the
Korean Mathematical Society CKMS

An le-module $M$ over a commutative ring $R$ is a complete lattice ordered additive monoid $(M, \leqslant, +)$ having the greatest element $e$ together with a module like action of $R$. This article characterizes the le-modules $_RM$ such that the pseudo-prime spectrum $X_M$ endowed with the Zariski topology is a Noetherian topological space. If the ring $R$ is Noetherian and the pseudo-prime radical of every submodule elements of $_{R}M$ coincides with its Zariski radical, then $X_{M}$ is a Noetherian topological space. Also we prove that if $R$ is Noetherian and for every submodule element $n$ of $M$ there is an ideal $I$ of $R$ such that $V(n) = V(Ie)$, then the topological space $X_{M}$ is spectral.

Keywords: Pseudo-prime submodule element, Zariski topology, topological le-module, Noetherian space, spectral space

MSC numbers: 54B35, 13C05, 13C99, 06F25