On prime fuzzy submodules and radicals (Q2710036)

Let \(R\) be a commutative ring with identity and let \(M\) be a unitary \(R\)-module. Let \(a\in M\) and let \(\lambda\in [0,1]\). A fuzzy subset \(a^\lambda\) of \(M\) is a fuzzy subset of \(M\) such that \(a^\lambda(x)=\lambda\) if \(x=a\) and \(a^\lambda(x)=0\) otherwise. Let \(\nu\) be a fuzzy subset of \(M\). Let \(\mu\) denote a fuzzy submodule of \(M\). Let \(\nu\) be a fuzzy submodule of \(M\) such that \(\nu\subseteq\mu\). Then the fuzzy residual of \(\nu\) in \(\mu\), written \(\nu:\mu\), is defined as follows: For all \(r\in R\), \((\nu:\mu) (r)=\sup\{\omega(r) \mid \omega \circ\mu \subseteq \nu\), \(\omega\) is a fuzzy ideal of \(R\}\), where \((\omega \circ \nu)(x)= \sup\{\omega(r) \wedge\mu (a)\mid x=ra\), \(r\in R\), \(a\in M\}\) for all \(x\in M\). The fuzzy submodule \(\nu\) is called prime (primary) in \(\mu\) if \(\nu\neq\mu\) and for \(r^\beta \subseteq R\) (the characteristic function of \(R)\) and \(m^\lambda \subseteq\mu\), \(r^\beta m^\lambda \subseteq\nu\) implies either \(m^\lambda \subseteq\nu\) or \(r^\beta \subseteq\nu: \mu\;\) \((r^\beta \subseteq \sqrt{\nu: \mu})\).NEWLINENEWLINENEWLINEThe authors characterize prime and primary fuzzy submodules of \(M\). They show that if \(\mu\) is a prime (primary) fuzzy submodule of \(M\), thenNEWLINENEWLINENEWLINE(i) the level set \(\mu^1\) is prime,NEWLINENEWLINENEWLINE(ii) \(\text{Im} (\mu) =\{t,1\}\) for some \(t\in [0,1)\), andNEWLINENEWLINENEWLINE(iii) \(\mu:M\) is a prime (primary) fuzzy ideal in \(R\).NEWLINENEWLINENEWLINEThey also show that if \(\text{Im} (\mu)= \{t,1\}\) form some \(t\in [0,1)\) and \(\mu^1\) is a prime (primary) fuzzy submodule of \(M\), then \(\mu\) is a prime (primary) fuzzy submodule of \(M\).NEWLINENEWLINENEWLINEThe authors then define the radical of \(\mu\) and prove some properties concerning it. They then turn their attention to the existence of minimal prime fuzzy submodules.