On prime fuzzy submodules (Q2710038)

Let \(R\) denote a commutative ring with identity, \(M\) a left \(R\)-module, and \(L\) a completely distributive lattice with least element 0 and greatest element 1. A fuzzy subset of a nonempty set \(X\) is a function of \(X\) into \(L\). In this paper, the authors define the concepts of strongly prime, weakly prime, and prime fuzzy submodules and give a characterization of them. Let \(I(R)\) denote the set of all fuzzy ideals of \(R\) and let \(S(M)\) denote the set of all fuzzy submodules of \(M\). Let \(A\) denote a non-constant fuzzy submodule of \(M\). Then \(A\) is called strongly prime if for all \(I\in I(R)\) and for all \(B\in S(M)\), \(I\circ B\subset A\) implies either \(B\subseteq A\) or \(I(r) \leq A(rm)\) for all \(r\in R\) and for all \(m\in M\), where \(I\circ B(m)=\vee \{I(r) \wedge B(x)\mid m=rx\}\). The authors then show that \(A\) is strongly prime if and only if \(A(0)=1\), \(\text{Im}(A)= \{1,t\}\), where \(t\) is prime, and \(A_*\) is prime, where \(A_*=\{m\in M\mid A(m)= A(0)\}\) if and only if \(r_sx_t\subseteq A\) implies either \(x_t\subseteq A\) or \((rm)_s\subseteq A\) for all \(m\in M\), where \(r_s\) and \(x_t\) denote fuzzy singletons.NEWLINENEWLINENEWLINENow let \(A\) be a fuzzy submodule of \(M\). Then \(A\) is called prime if \(A(rx)> A(x)\) implies that \(A(rx)\leq A(rm)\) for all \(r\in R\) and \(x,m\in M\). The authors show that \(A\) is prime if and only if the level sets \(A_t\) are prime submodules of \(M\) for all \(t\in [0,A(0)]\). \(A\) is called weakly prime if \(A(rx)= A(0)\) implies that \(A(x)=A(0)\) or \(A(rm)=A(0)\) for all \(m\in M\). The authors show that \(A\) is weakly prime if and only if \(A(rx) <A(0)\) implies that \(A(rm)<A(0)\) for all \(m\in M\) such that \(A(m)<A(0)\).