G r - ( 2 , n ) -ideals in graded commutative rings

Khaldoun Al-Zoubi; Shatha Alghueiri; Ece Y. Celikel

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 2, page 129-138
  • ISSN: 0010-2628

Abstract

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Let G be a group with identity e and let R be a G -graded ring. In this paper, we introduce and study the concept of graded ( 2 , n ) -ideals of R . A proper graded ideal I of R is called a graded ( 2 , n ) -ideal of R if whenever r s t ∈ I where r , s , t ∈ h ( R ) , then either r t ∈ I or r s ∈ G r ( 0 ) or s t ∈ G r ( 0 ) . We introduce several results concerning g r - ( 2 , n ) -ideals. For example, we give a characterization of graded ( 2 , n ) -ideals and their homogeneous components. Also, the relations between graded ( 2 , n ) -ideals and others that already exist, namely, the graded prime ideals, the graded 2-absorbing primary ideals, and the graded n -ideals are studied.

How to cite

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Al-Zoubi, Khaldoun, Alghueiri, Shatha, and Celikel, Ece Y.. "$Gr$-$(2,n)$-ideals in graded commutative rings." Commentationes Mathematicae Universitatis Carolinae 61.2 (2020): 129-138. <http://eudml.org/doc/297189>.

@article{Al2020,
abstract = {Let $G$ be a group with identity $e$ and let $R$ be a $G$-graded ring. In this paper, we introduce and study the concept of graded $(2,n)$-ideals of $R$. A proper graded ideal $I$ of $R$ is called a graded $(2,n)$-ideal of $R$ if whenever $rst\in I$ where $r,s,t\in h(R)$, then either $rt\in I$ or $rs\in Gr(0)$ or $st\in Gr(0)$. We introduce several results concerning $gr$-$(2,n)$-ideals. For example, we give a characterization of graded $(2,n)$-ideals and their homogeneous components. Also, the relations between graded $(2,n)$-ideals and others that already exist, namely, the graded prime ideals, the graded 2-absorbing primary ideals, and the graded $n$-ideals are studied.},
author = {Al-Zoubi, Khaldoun, Alghueiri, Shatha, Celikel, Ece Y.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$gr$-$(2,n)$-ideals; $gr$-$2$-absorbing primary ideals; $gr$-prime ideal},
language = {eng},
number = {2},
pages = {129-138},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {$Gr$-$(2,n)$-ideals in graded commutative rings},
url = {http://eudml.org/doc/297189},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Al-Zoubi, Khaldoun
AU - Alghueiri, Shatha
AU - Celikel, Ece Y.
TI - $Gr$-$(2,n)$-ideals in graded commutative rings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 2
SP - 129
EP - 138
AB - Let $G$ be a group with identity $e$ and let $R$ be a $G$-graded ring. In this paper, we introduce and study the concept of graded $(2,n)$-ideals of $R$. A proper graded ideal $I$ of $R$ is called a graded $(2,n)$-ideal of $R$ if whenever $rst\in I$ where $r,s,t\in h(R)$, then either $rt\in I$ or $rs\in Gr(0)$ or $st\in Gr(0)$. We introduce several results concerning $gr$-$(2,n)$-ideals. For example, we give a characterization of graded $(2,n)$-ideals and their homogeneous components. Also, the relations between graded $(2,n)$-ideals and others that already exist, namely, the graded prime ideals, the graded 2-absorbing primary ideals, and the graded $n$-ideals are studied.
LA - eng
KW - $gr$-$(2,n)$-ideals; $gr$-$2$-absorbing primary ideals; $gr$-prime ideal
UR - http://eudml.org/doc/297189
ER -

References

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  9. Năstăsescu C., van Oystaeyen F., Graded Ring Theory, North-Holland Publishing, Amsterdam, 1982. MR0676974
  10. Năstăsescu C., van Oystaeyen F., Methods of Graded Rings, Lecture Notes in Mathematics, 1836, Springer, Berlin, 2004. MR2046303
  11. Refai M., Al-Zoubi K., On graded primary ideals, Turkish J. Math. 28 (2004), no. 3, 217–229. MR2095827
  12. Refai M., Hailat M., Obiedat S., Graded radicals and graded prime spectra, Far East J. Math. Sci. (FJMS) Special Volume, Part I (2000), 59–73. MR1761071
  13. Tamekkante M., Bouba El M., 10.1142/S0219498819501032, J. Algebra Appl. 18 (2019), no. 6, 1950103, 12 pages. MR3954657DOI10.1142/S0219498819501032

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