Generalizations of prime ideals. (English) Zbl 1140.13005

Anderson, D. D.; Bataineh, Malik

Commun. Algebra 36, No. 2, 686-696 (2008).

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.

Reviewer: Septimiu Crivei (Cluj-Napoca)

MSC:

13A15

Ideals and multiplicative ideal theory in commutative rings

Keywords:

almost prime ideal; prime ideal; \(\phi\)-prime ideal; weakly prime ideal

References:

[1]	Agargün A. G., Comm. Algebra 27 pp 1967– (1999) · Zbl 0947.13013
[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
[5]	Bhatwadekar S. M., Comm. Algebra 33 pp 43– (2005) · Zbl 1072.13003
[6]	Galovich S., Math. Mag. 51 pp 276– (1978) · Zbl 0407.13013
[7]	Hedstrom J. R., Pacific J. Math. 75 pp 137– (1978) · Zbl 0368.13002
[8]	Kaplansky I., Commutative Rings. () (1974)