Graded weakly 1-absorbing prime ideals

Ìnsal Tekir; Suat  Koç; Rashid Abu-Dawwas; Eda YÄ±ldÄ±z

doi:10.56754/0719-0646.2402.0291

DOI:

https://doi.org/10.56754/0719-0646.2402.0291

Abstract

In this paper, we introduce and study graded weakly 1-absorbing prime ideals in graded commutative rings. Let \(G\) be a group and \(R\) be a \(G\)-graded commutative ring with a nonzero identity \(1\neq0\). A proper graded ideal \(P\) of \(R\) is called a graded weakly 1-absorbing prime ideal if for each nonunits \(x,y,z\in h(R)\) with \(0\neq xyz\in P\), then either \(xy\in P\) or \(z\in P\). We give many properties and characterizations of graded weakly 1-absorbing prime ideals. Moreover, we investigate weakly 1-absorbing prime ideals under homomorphism, in factor ring, in rings of fractions, in idealization.

Keywords

graded ideal , 1-absorbing prime ideal , weakly 1-absorbing prime ideal , graded weakly 1-absorbing prime ideal

Ìnsal Tekir Department of Mathematics, Marmara University, Istanbul, Turkey.
Suat Koç Department of Mathematics, Istanbul Medeniyet University, Istanbul, Turkey.
Rashid Abu-Dawwas Department of Mathematics, Yarmouk University, Jordan.
Eda YÄ±ldÄ±z Department of Mathematics, Yildiz Technical University, Istanbul, Turkey.

Pages: 291–305
Date Published: 2022-08-22
Vol. 24 No. 2 (2022)

R. Abu-Dawwas, E. YÄ±ldÄ±z, Ì. Tekir and S. Koç, “On graded 1-absorbing prime ideals‘”, São Paulo J. Math. Sci., vol. 15, no. 1, pp. 450–462, 2021.

A. Assarrar, N. Mahdou, Ì. Tekir and S. Koç, “On graded coherent-like properties in trivial ring extensions”, Boll. Unione Mat. Ital., vol. 15, pp. 437–449, 2022.

K. Al-Zoubi, R. Abu-Dawwas and S. C Ì§eken, “On graded 2-absorbing and graded weakly 2-absorbing ideals”, Hacet. J. Math. Stat., vol. 48, no. 3, pp. 724–731, 2019.

D. D. Anderson and E. Smith, “Weakly prime ideals”, Houston J. Math, vol. 29, no. 4, pp. 831–840, 2003.

S. E. Atani, “On graded weakly prime ideals”, Turkish J. Math., vol. 30, no. 4, pp. 351–358, 2006.

A. Badawi and E. Y. Celikel, “On 1-absorbing primary ideals of commutative rings”, J. Algebra Appl., vol. 19, no. 6, 12 pages, 2020.

M. Bataineh and R. Abu-Dawwas, “On graded 2-prime ideals”, Mathematics, vol. 9, no. 5, Paper No. 493, 10 pages, 2021.

M. Bataineh and R. Abu-Dawwas, “Graded 1-absorbing primary ideals”, submitted.

F. Farzalipour and P. Ghiasvand, “On the union of graded prime submodules”, Thai J. Math., vol. 9, no. 1, pp. 49–55, 2011.

A. Jaber, M. Bataineh and H. Khashan, “Almost graded prime ideals”, J. Math. Stat., vol. 4, no. 4, pp. 231–235, 2008.

S. Koç, Ì. Tekir and E. Yildiz, “On weakly 1-absorbing prime ideals”, Ricerche di Matematica, pp. 1–16, 2021.

S. R. Naghani and H. F. Moghimi, “On graded 2-absorbing and graded weakly 2-absorbing ideals of a commutative ring”, Çankaya University Journal of Science and Engineering, vol. 13, no. 2, pp. 11–17, 2016.

C. NÄƒstÄƒsescu and F. Van Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics, vol. 1836, Berlin: Springer-Verlag , 2004.

M. Refai and K. Al-Zoubi, “On graded primary ideals”, Turkish J. Math., vol. 28, no. 3, pp. 217–229, 2004.

M. Refai, M. Hailat and S. Obiedat, “Graded radicals and graded prime spectra”, Far East J. Math. Sci., Special Volume, Part I, pp. 59–73, 2000.

F. Soheilnia and A. Y. Darani, “On graded 2-absorbing and graded weakly 2-absorbing primary ideals”, Kyungpook Math. J., vol. 57, no. 4, pp. 559–580, 2017.

R. N. Uregen, Ì. Tekir, K. P. Shum and S. Koç, “On graded 2-absorbing quasi primary ideals”, Southeast Asian Bull. Math., vol. 43, no. 4, pp. 601–613, 2019.

J. Van Geel and F. Van Oystaeyen, “About graded fields”, Nederl. Akad. Wetensch. Indag. Math., vol. 43, no. 3, pp. 273–286, 1981.

A. Yassine, M. J. Nikmehr and R. Nikandish, “On 1-absorbing prime ideals of commutative rings”, J. Algebra Appl., vol. 20, no. 10, Paper No. 2150175, 12 pages, 2021.