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Generalizations of primal ideals in commutative rings. (English) Zbl 1299.13004

Authors’ abstract: Let \(R\) be a commutative ring with identity. Let \(\phi:\mathcal I(R)\to\mathcal I(R)\cup\{\emptyset\}\) be a function where \(\mathcal I(R)\) denotes the set of all ideals of \(R\). Let \(I\) be an ideal of \(R\). An element \(a\in R\) is called \(\phi\)-prime to \(I\) if \(ra\in I-\phi(I)\) (with \(r\in R\)) implies that \(r\in I\). We denote by \(S_\phi(I)\) the set of all elements of \(R\) that are not \(\phi\)-prime to \(I\). \(I\) is called a \(\phi\)-primal ideal of \(R\) if the set \(P:= S_\phi(I)\cup\phi(I)\) forms an ideal of \(R\). So if we take \(\phi_{\emptyset}(Q)=\emptyset\) (resp., \(\phi_0(Q)=0\)), a \(\phi\)-primal ideal is primal (resp., weakly primal). We study the properties of several generalizations of primal ideals of \(R\).

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
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