Generalizations of prime ideals. (English) Zbl 1140.13005

Throughout \(R\) is a commutative ring with identity. The authors introduce the following generalization of the notion of prime ideal. Denote by \(\mathcal{I}(R)\) the set of ideals of \(R\). Let \(\phi:\mathcal{I}(R)\to \mathcal{I}(R)\cup \{\emptyset\}\) be a function. A proper ideal \(I\) of \(R\) is called \(\phi\)-prime if \(a,b\in R\) with \(ab\in I-\phi(I)\) implies \(a\in I\) or \(b\in I\). Taking \(\phi_{\emptyset}(J)=\emptyset\) (respectively \(\phi_0(J)=0\), \(\phi_2(J)=J^2\)) a \(\phi_{\emptyset}\)-prime (respectively \(\phi_0\)-prime, \(\phi_2\)-prime) ideal is just a prime (respectively weakly prime, almost prime) ideal. The authors establish various properties of \(\phi\)-prime ideals, analogs of those of prime ideals.


13A15 Ideals and multiplicative ideal theory in commutative rings
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