## 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.

### MSC:

 13A15 Ideals and multiplicative ideal theory in commutative rings
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### References:

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