An extension of $z$-ideals and $z^\circ$-ideals
| Autor: | Sajad Nazari, Mehdi Badie, Ali Rezaei Aliabad |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Statistics and Probability
Ring (mathematics) Zariski topology Matematik Algebra and Number Theory Mathematics::Commutative Algebra Prime ideal 010102 general mathematics Spectrum (functional analysis) Semiprime 010103 numerical & computational mathematics Commutative ring 01 natural sciences Combinatorics Annihilator $z$-ideal $z^\circ$-ideal strong $z$-ideal strong $z^\circ$-ideal prime ideal semiprime ideal Zariski topology Hilbert ideal rings of continuous functions Geometry and Topology Ideal (ring theory) 0101 mathematics Analysis Mathematics |
| Zdroj: | Volume: 49, Issue: 1 254-272 Hacettepe Journal of Mathematics and Statistics |
| ISSN: | 2651-477X |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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