Relative \(z\)-ideals in commutative rings. (English) Zbl 1264.13003

Aliabad, A. R.; Azarpanah, F.; Taherifar, A.

Commun. Algebra 41, No. 1, 325-341 (2013).

This paper deals with the fundamental properties of relative \(z\)-ideal (briefly re\(z\)-ideal) which has been introduced by Aliabad and Azarpanah and Taherifar. This class of ideals is a special kind of generalized form of the class of \(z\)-ideals. After giving a clear and informative introduction, the authors not only characterize re\(z\)-ideals but also offer some examples of re\(z\)-ideal in Section 2. Among others, they show that whenever an ideal \(I\) is a re\(z\)-ideal, then it is also a \(z_K\)-ideal for some ideal \(K\) containing \(I\) properly. In Proposition 2.8, they show that if \(I\) is an ideal in a semisimple ring \(R\) with Ann(\(I)\neq (0)\), then \(I\) is a re\(z\)-ideal. Moreover, they show that if \(P\) and \(Q\) are prime ideals in a ring \(R\) and if \(P \cap Q\) is re\(z\)-ideal, then either \(P\) or \(Q\) is a \(z\)-ideal (see Proposition 2.10). In Section 3, the authors consider large and small re\(z\)-ideals. More specifically, maximal and minimal \(z\)-factors of \(I\) are investigated for an ideal \(J\) in a ring \(R\), maximal \(z_J\)-ideals in \(R\) and for a given ideal \(I\) in \(R\). Indeed, nontrivial maximal \(z_J\)-ideals are characterized as the maximal elements of the set of prime \(z\)-ideals that do not contain \(J\) and nontrivial maximal \(z_J\)-ideals contained in \(J\) are precisely the intersection of such prime \(z\)-ideals with \(J\). The authors characterize the largest \(z\)-factor of ideals if it exists. Furthermore, they show that the largest \(z\)-factor of a semiprime re\(z\)-ideal \(I\) is the intersection of those minimal prime ideals over \(I\) which are not \(z\)-ideals. Section 4 is dedicated to the investigation of the ring \(C(X)\) via re\(z\)-ideals by which they obtain some nice results concerning the algebraic properties of \(C(X)\) and topological properties of \(X\). Among others, they show that the sum of a re\(z\)-ideal and a convex (absolutely convex) ideal is convex (absolutely convex) iff \(X\) is an \(F\)-space.

Reviewer: Saeid Jafari (Copenhagen)

MSC:

13A15	Ideals and multiplicative ideal theory in commutative rings
54C40	Algebraic properties of function spaces in general topology

Keywords:

convex and absolutely convex ideal; \(z\)-ideal; \(F\)-space; re\(z\)-ideal; \(z\)-factor; \(z_J\)-ideal

References:

[1]	Aliabad A. R., z{\(\deg\)}-ideals in C(X) (1996)
[2]	Azarpanah F., Proc. Amer. Math. Soc. 125 pp 2149– (1997) · Zbl 0867.54023 · doi:10.1090/S0002-9939-97-04086-0
[3]	Azarpanah F., Period. Math. Hungar. 31 pp 105– (1995) · Zbl 0869.54021 · doi:10.1007/BF01876485
[4]	Azarpanah F., Acta Math. Sin. (Engl. Ser.) 23 pp 989– (2007) · Zbl 1186.54021 · doi:10.1007/s10114-005-0738-7
[5]	Azarpanah F., Topology Appl. 156 pp 1711– (2009) · Zbl 1167.54005 · doi:10.1016/j.topol.2009.02.002
[6]	Birkhoff G., Lattice theory. Amer Math. Soc. (1948)
[7]	Cornish W. H., Abelian Ricart-semirings (1970)
[8]	Engelking R., General Topology (1977)
[9]	Gillman L., Rings of Continuous Functions (1976) · Zbl 0339.46018
[10]	Huijsmans C. B., Indag. Math. 42. Proc. Netherl. Acad. Sc. A 83 pp 391– (1980)
[11]	Karamzadeh O. A. S., Proc. Amer. Math. Soc. 93 pp 179– (1985)
[12]	McConnel J. C., Noncommutative Noetherian Rings (1987)
[13]	Mason G., Canad. Math. Bull. 23 (4) pp 437– (1980) · Zbl 0455.54010 · doi:10.4153/CMB-1980-064-3
[14]	Mason G., J. Algebra 26 pp 280– (1973) · Zbl 0262.13003 · doi:10.1016/0021-8693(73)90024-0
[15]	Mulero M. A., Fund. Math. 149 pp 55– (1996)
[16]	Rudd D., Michigan Math. J. 17 pp 139– (1970) · Zbl 0194.44403 · doi:10.1307/mmj/1029000423