Relative z-Ideals in Commutative Rings

Abstract

If I and J are two ideals in a ring R, we call I a z_J-ideal if M_a ∩ J ⊆ I, ∀ a ∈ I, where M_a is the intersection of all maximal ideals containing a. Whenever J ⊈ I and I is a z_J-ideal, we say that I is a relative z-ideal or briefly a rez-ideal, and we call J a z-factor of I. If I is an ideal in a semisimple ring and Ann(I) ≠ (0), we have shown that I is a rez-ideal and the converse is also true for each finitely generated ideal in C(X). Small and large rez-ideals and z-factors are investigated and the largest z-factor of a given semiprime ideal and maximal z_J-ideals for a given ideal J are obtained. Minimal z-factors of a given ideal in a reduced ring are also characterized and it turns out that minimal z-factors of a given ideal I in C(X) are precisely the minimal ideals of C(X) which are not contained in I. Finally, we show that the space X is an F-space if and only if the sum of every rez-ideal and every convex ideal in C(X) is convex, and spaces X for which the intersection of every two semiprime rez-ideals in C(X) is a rez-ideal are characterized.

Key Words:

• Convex and absolutely convex ideal
• F-space
• rez-ideal
• z-factor
• z _J -ideal

2010 Mathematics Subject Classification:

• 13A15
• 54C40

Notes

Communicated by S. Bazzoni.