On SZ°-Ideals in Polynomial Rings

Abstract

The theory of z-ideals and z°-ideals, especially as pertaining to the ideal theory of C(X), the ring of continuous functions on a completely regular Hausdorff space X, has been attended to during the recent years; see Gillman and Jerison [9], Mason [18], and Azarpanah et al. [4]. In this article we will consider the theory of z°-ideals as applied to the rings of polynomials over a commutative ring with identity. We introduce and study sz°-ideals (an ideal I of a ring is called sz°-ideal, if whenever S is a finite subset of I, then the intersection of all minimal prime ideals containing S is in I). In addition, we will pay attention to several annihilator conditions and find some new results. Finally, we use the two examples that appeared in Henriksen and Jerison [10] and Huckaba [12], to answer some natural questions that might arise in the literature.

Key Words:

• Annihilator conditions
• Annihilator map
• Coefficient map
• Minimal prime ideal
• Polynomial ring
• Reduced ring
• sz°-Ideal
• z°-Ideal

2000 Mathematics Subject Classification:

• 11C08
• 13A15
• 13B25
• 13F20
• 54C40

ACKNOWLEDGMENTS

We are very grateful to the referee, whose valuable suggestions and comments substantially improved an earlier version of this article. Thanks are also due to professor O. A. S. Karamzadeh for his advice and encouragements during the preparation of the revised version.

Notes

Communicated by S. Sehgal.