CLASSICAL PRIME AND 2-ABSORBING L-SUBMODULES

Abstract

Let $L$ be a complete lattice. Let $R$ be a commutative ring, $M$ an $R$-module and $\nu$ an $L$-submodule of $M$. $\nu$ is called a classical prime $L$-submodule of $M$ if for any $L$-fuzzy points $a_r, b_s\in L^R$ and $x_t\in L^M$ ($a, b\in R$, $x\in M$ and $r, s, t\in L$), $a_rb_sx_t\in \nu$ implies that either $a_rx_t\in \nu$ or $b_sx_t\in \nu$. Assume that $\nu$ is an $L$-submodule of $mmu\in L(M)$. We say that $\nu$ is a $2$-absorbing $L$-submodule of $\mu$ if for any $L$-fuzzy points $a_r, b_s\in L^R$ and $x_t\in L^M$ ($a, b\in R$, $x\in M$ and $r, s, t\in L$), $a_rb_sx_t\in \nu$ implies that $a_rb_s\mu\subseteq \nu$ or $a_rx_t\in \nu$ or $b_sx_t\in \nu$. In this case every prime $L$-submodule of $M$ is a classical prime $L$-submodule, and every classical prime $L$-submodule is a $2$-absorbing $L$-submodule. In this paper we give some basic results concerning these classes of $L$-submodules. Finally we topologize $L-Cl.Spec(M)$, the set of all classical prime $L$-submodules of $M$, with Zariski topology.