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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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[1] Agarg√ľn 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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