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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2020 / Cilt: 49 - Sayı: 1

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

Ali Rezaei ALİABAD
Mehdi BADİE
Sajad NAZARİ
Sayfa
254–272
Yayın Tarihi
2020-02-06
DOI
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

This comprehensive look helps to recognize the resemblances and differences of known concepts better.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:Let $R$ be a commutative ring, $Y\subseteq Spec(R)$ and $ h_Y(S)=\{P\in Y:S\subseteq P \}$, for every $S\subseteq R$.
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Temel Bulgular & Veri:An ideal $I$ is said to be an $\mathcal{H}_Y$-ideal whenever it follows from $h_Y(a)\subseteq h_Y(b)$ and $a\in I$ that $b\in I$.
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Sonuç & Katkı:This comprehensive look helps to recognize the resemblances and differences of known concepts better.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:$z$-ideal · $z^\circ$-ideal · strong $z$-ideal

Abstract

Let $R$ be a commutative ring, $Y\subseteq Spec(R)$ and $ h_Y(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 $h_Y(a)\subseteq h_Y(b)$ 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 (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 (d-ideal), sz-ideal and $sz^{\circ}$-ideal ($\xi$-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.

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Bu Makaleye Atıf Yap

ALİABAD, A. R., BADİE, M., NAZARİ, S. (2020). An extension of $z$-ideals and $z^\circ$-ideals. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 1, 254–272. https://ajindex.com/articles/669310

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