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

#### Ali Rezaei ALİABAD [1] , Mehdi BADİE [2] , Sajad NAZARİ [3]

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.
$z$-ideal, $z^\circ$-ideal, strong $z$-ideal, strong $z^\circ$-ideal, prime ideal, semiprime ideal, Zariski topology, Hilbert ideal, rings of continuous functions
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Primary Language en Mathematics Mathematics Orcid: 0000-0003-1293-3652Author: Ali Rezaei ALİABAD Institution: Shahid Chamran University of AhvazCountry: Iran Orcid: 0000-0003-1114-3130Author: Mehdi BADİE (Primary Author)Institution: Jundi-Shapur University of TechnologyCountry: Iran Orcid: 0000-0002-4295-2435Author: Sajad NAZARİ Institution: Shahid Chamran University of AhvazCountry: Iran Publication Date : February 6, 2020
 Bibtex @research article { hujms455030, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {254 - 272}, doi = {10.15672/hujms.455030}, title = {An extension of \$z\$-ideals and \$z\^\\circ\$-ideals}, key = {cite}, author = {ALİABAD, Ali Rezaei and BADİE, Mehdi and NAZARİ, Sajad} } APA ALİABAD, A , BADİE, M , NAZARİ, S . (2020). An extension of $z$-ideals and $z^\circ$-ideals. Hacettepe Journal of Mathematics and Statistics , 49 (1) , 254-272 . DOI: 10.15672/hujms.455030 MLA ALİABAD, A , BADİE, M , NAZARİ, S . "An extension of $z$-ideals and $z^\circ$-ideals". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 254-272 Chicago ALİABAD, A , BADİE, M , NAZARİ, S . "An extension of $z$-ideals and $z^\circ$-ideals". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 254-272 RIS TY - JOUR T1 - An extension of $z$-ideals and $z^\circ$-ideals AU - Ali Rezaei ALİABAD , Mehdi BADİE , Sajad NAZARİ Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.455030 DO - 10.15672/hujms.455030 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 254 EP - 272 VL - 49 IS - 1 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.455030 UR - https://doi.org/10.15672/hujms.455030 Y2 - 2018 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics An extension of $z$-ideals and $z^\circ$-ideals %A Ali Rezaei ALİABAD , Mehdi BADİE , Sajad NAZARİ %T An extension of $z$-ideals and $z^\circ$-ideals %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 1 %R doi: 10.15672/hujms.455030 %U 10.15672/hujms.455030 ISNAD ALİABAD, Ali Rezaei , BADİE, Mehdi , NAZARİ, Sajad . "An extension of $z$-ideals and $z^\circ$-ideals". Hacettepe Journal of Mathematics and Statistics 49 / 1 (February 2020): 254-272 . https://doi.org/10.15672/hujms.455030 AMA ALİABAD A , BADİE M , NAZARİ S . An extension of $z$-ideals and $z^\circ$-ideals. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 254-272. Vancouver ALİABAD A , BADİE M , NAZARİ S . An extension of $z$-ideals and $z^\circ$-ideals. Hacettepe Journal of Mathematics and Statistics. 2020; 49(1): 272-254.

Authors of the Article
[1]
[3]