An extension of $z$-ideals and $z^\circ$-ideals

Authors : Ali Rezaei ALİABAD, Mehdi BADİE, Sajad NAZARİ

Pages : 254-272

Doi:10.15672/hujms.455030

View : 18 | Download : 10

Publication Date : 2020-02-06

Article Type : Research Paper

Abstract :Let $R$ be a commutative ring, $Y\subseteq Specinsert ignore into journalissuearticles values(R);$ and $ h_Yinsert ignore into journalissuearticles values(S);=\{P\in Y:S\subseteq P \}$, for every $S\subseteq R$. An ideal $I$ is said to be an $\mathcal{H}_Y$-ideal whenever it follows from $h_Yinsert ignore into journalissuearticles values(a);\subseteq h_Yinsert ignore into journalissuearticles values(b);$ and $a\in I$ that $b\in I$. A strong  $\mathcal{H}_Y$-ideal is defined in the same way by replacing an arbitrary finite set $F$ instead of the element $a$. In this paper these two classes of ideals insert ignore into journalissuearticles values(which are based on the spectrum of the ring $R$ and are a generalization of the well-known concepts semiprime ideal, z-ideal, $z^{\circ}$-ideal insert ignore into journalissuearticles values(d-ideal);, sz-ideal and $sz^{\circ}$-ideal insert ignore into journalissuearticles values($\xi$-ideal);); are studied. We show that the most important results about these concepts, Zariski topology`, annihilator` and etc can be extended in such a way that the corresponding consequences seems to be trivial and useless. This comprehensive look helps to recognize the resemblances and differences of known concepts better.
Keywords : z ideal, z^circ ideal, strong z ideal, strong z^circ ideal, prime ideal, semiprime ideal, Zariski topology, Hilbert ideal, rings of continuous functions