On SZ °-Ideals in Polynomial Rings

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 Gillman , L. , Jerison , M. ( 1976 ). Rings of Continuous Functions . New York : Springer Verlag . [Google Scholar]], Mason [18 Mason , G. ( 1989 ). Prime ideals and quotient rings of reduced rings . Math. Japan 34 : 941 – 956 . [Google Scholar]], and Azarpanah et al. [4 Azarpanah , F. , Karamzadeh , O. A. S. , Rezaei Aliabad , A. ( 2000 ). On ideals consisting entirely of zerodivisor . Comm. Algebra 28 ( 2 ): 1061 – 1073 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. 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 Henriksen , M. , Jerison , M. ( 1965 ). The space of minimal prime ideals of a commutative ring . Trans. Amer. Math. Soc. 115 : 110 – 130 .[Crossref], [Web of Science ®] , [Google Scholar]] and Huckaba [12 Huckaba , J. A. ( 1988 ). Commutative Rings with Zero Divisors . Marcel-Dekker Inc . [Google Scholar]], to answer some natural questions that might arise in the literature.