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On fuzzy essential ideals of rings. (English) Zbl 1165.16002

The authors examine essential ideals of rings. Let \(R\) denote a noncommutative ring with identity. Let \(\mu\) be a fuzzy subset of \(R\), i.e., a function of \(R\) into the closed interval \([0,1]\). For all \(t\in[0,1]\), let \(\mu_t=\{x\in R\mid\mu(x)\geq t\}\). Let \(l(S)\) denote the left annihilator of \(S\), where \(S\) is a subset of \(R\). The ‘left fuzzy annihilator’ of \(\mu\), denoted \(l(\mu)\), is defined as follows: \(\forall x\in R\), \[ l(\mu)(x)=\begin{cases}\bigvee\{t\in[0,1]\mid t\in\text{Im}(\mu)\}\text{ if }x\in l(\mu_t),\\ 0\text{ otherwise}.\end{cases} \] The right annihilator of \(\mu\) is defined similarly. A left ideal \(A\) of \(R\) is called ‘essential’, denoted \(A\subseteq_eR\), if for every nonzero left ideal \(B\) of \(R\), \(A\cap B\neq\{0\}\). Let \(\mu\) and \(\sigma\) be nonzero fuzzy left ideals of \(R\) such that \(\mu\subseteq\sigma\). Then \(\mu\) is called a ‘fuzzy essential left ideal’ of \(\sigma\), denoted \(\mu\subseteq_e\sigma\), if for every nonzero fuzzy left ideal \(\theta\) of \(R\) with \(\theta\subseteq\sigma\), there exists \(x\in R\), \(x\neq 0\), such that \(x_t\subseteq\mu\) and \(x_t\subseteq\theta\) for all \(t\in(0,\sigma(0)]\), where \(x_t(y)=t\) if \(y=x\) and \(x_t(y)=0\) otherwise. Let \(Z_f(R)=\{r\in R\mid\mu_r=\chi_0\) for some fuzzy essential left ideal \(\mu\) of \(R\}\).
Among other results the authors prove the following: Let \(\mu,\nu\), and \(\sigma\) be nonzero fuzzy left ideals of \(R\) such that \(\mu\subseteq\nu\subseteq\sigma\). Then \(\mu\subseteq_e\sigma\) if and only if \(\mu\subseteq_e\nu\subseteq_e\sigma\). – If the fuzzy essential left ideals of \(R\) satisfy the supremum property, then \(Z_f(R)\) is a left ideal of \(R\). – \(r\in Z_f(R)\) if and only if \(l(\chi_{(r)})\subseteq_eR\).

MSC:

16D25 Ideals in associative algebras
16Y99 Generalizations
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