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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:
 [1] Agargün A. G., Comm. Algebra 27 pp 1967– (1999) · Zbl 0947.13013 · doi:10.1080/00927879908826543 [2] Anderson D. D., Houston J. Math. 29 pp 831– (2003) [3] Atani E. S., Georgian Math. J. 12 pp 423– (2005) [4] Badawi A., Bull. Austral. Math. Soc. 75 pp 417– (2007) · Zbl 1120.13004 · doi:10.1017/S0004972700039344 [5] Bhatwadekar S. M., Comm. Algebra 33 pp 43– (2005) · Zbl 1072.13003 · doi:10.1081/AGB-200034161 [6] Galovich S., Math. Mag. 51 pp 276– (1978) · Zbl 0407.13013 · doi:10.2307/2690246 [7] Hedstrom J. R., Pacific J. Math. 75 pp 137– (1978) · Zbl 0368.13002 · doi:10.2140/pjm.1978.75.137 [8] Kaplansky I., Commutative Rings. () (1974)
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