On Weakly 1-Absorbing Primary Ideals of Commutative Semirings

Mohammad Saleh; Ibaa Muraa

doi:10.33434/cams.1195074

EN

On Weakly 1-Absorbing Primary Ideals of Commutative Semirings

Abstract

Let $R$ be a commutative semiring with $ 1 \neq0$. In this paper, we study the concept of weakly 1-absorbing primary ideal which is a generalization of 1-absorbing ideal over commutative semirings . A proper ideal $I$ of a semiring $R$ is called a weakly 1-absorbing primary ideal if whenever nonunit elements $a,b,c \in R$ and $0 \neq abc \in I$, then $ab \in I $, or $c \in \sqrt{I}$. A number of results concerning weakly 1-absorbing primary ideals and examples of weakly 1-absorbing primary ideals are given. An ideal is called a subtractive ideal $I$ of a semiring $R$ is an ideal such that if $ x,x+y\in I$, then $ y\in I$. Subtractive ideals or k-ideals are helpful in proving in many results related to ideal theory over semirings.

Keywords

prime ideal , weakly primary , 1-absorbing primary ideal , 2-absorbing primary ideal , weakly 1-absorbing primary ideal , weakly 2-absorbing primary ideal , weakly primary ideal , weakly prime ideal

References

[1] D. D. Anderson, Some remarks on multiplication ideals, Math. Japon, 25 (1980), 463-469.
[2] D. D. Anderson, E. Smith, Weakly prime ideals, Houston J. Math., 229 (4), 831-840.
[3] R. E. Atani, The ideal theory in quotients of commutative semirings, Glas. Math., 42 (2007), 301-308.
[4] S. E. Atani, R. E. Atani, Ideal theory in commutative semirings, Bu. Acad. Stiinte Repub. Mold. Mat., 2 (2009), 14-23.
[5] S. E. Atani, R. E. Atani, Some remarks on partitioning semirings, An. St. Univ. Ovidius Constants, 18 (2010), 49-62.
[6] M. F. Atiyah, I. G. Mac Donalel, An Intoduction to Commutative Algebra, Addison-Wesley Publishing Company, 1969.
[7] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Aust. Math. Soc., 75(3) (2007), 417-429.
[8] A. Badawi, E. Y. Celikel, On weakly 1-absorbing primary ideals of commutative rings, Algebra Colloq., 29(2), 189-202.
[9] A. Badawi, E. Y. Celikel, On 1-absorbing primary ideals of a commutative ring, J. Algebra Its Appl., 19(6), 050111.
[10] S. Darani, On 2-absorbing and weakly 2-absorbing ideals of commutative semirings, Kyungpook Math. J., 52 (2012), 91-97.

[11] J. N. Ghaudhari, V. Gupta, Prime ideals in semirings, Bull. Malaysian Math. Sci. Soc., 34 (2011), 415-421.
[12] J. N. Ghaudhari, 2-absorbing ideals in semirings, Int. J. Algebra, 6 (2012), 265-270.
[13] J. S. Golan, Semirings and Their Applications, Kluwer Acadimic Publisher’s, Dordrecht, 1999.
[14] M. Henricksen, Ideals in semirings with commutative addition, American Mathematics Society, 6 (1958), 3-12.
[15] P. Kumar. M. K. Dubey, P. Sarohe, On 2-absorbing primary ideals in commutative semirings, Eur. J. Pure Appl. Math., 9 (2016), 186-195.
[16] P. Nasehpour, Some remarks on semirings and their ideals, Asian-Eur. J. Math., 12(7), 2050002.
[17] R. J. Nezhad, A note on divided ideals, Pure Math. A, 22, 61-64.
[18] L. Sawalmeh, M. Saleh, On 2-absorbing ideals of commutative semirings, J. Algebra Its Appl., To appear.
[19] D. A. Smith, On semigroups, semirings and rings of quotients, J. Sci. Hirshimo Univ. Ser. Math., 2 (1966), 123-130.
[20] H. S. Vandiver, Note on a simple of algebra in which the cancellation law of addition dose not hold, Bull. Amer. Math. Soc., 40, 914-920.
[21] O. Zariski, P. Samuel, Commutative Algebra, V.I. Princeton, 1958.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Mohammad Saleh ^*
0000-0002-4254-2540
Palestine

Ibaa Muraa This is me
Palestine

Publication Date

December 30, 2022

Submission Date

October 26, 2022

Acceptance Date

December 29, 2022

Published in Issue

Year 2022 Volume: 5 Number: 4

DOI

https://doi.org/10.33434/cams.1195074

IZ

https://izlik.org/JA95YS45YF

APA

Saleh, M., & Muraa, I. (2022). On Weakly 1-Absorbing Primary Ideals of Commutative Semirings. Communications in Advanced Mathematical Sciences, 5(4), 199-208. https://doi.org/10.33434/cams.1195074

AMA

1.Saleh M, Muraa I. On Weakly 1-Absorbing Primary Ideals of Commutative Semirings. Communications in Advanced Mathematical Sciences. 2022;5(4):199-208. doi:10.33434/cams.1195074

Chicago

Saleh, Mohammad, and Ibaa Muraa. 2022. “On Weakly 1-Absorbing Primary Ideals of Commutative Semirings”. Communications in Advanced Mathematical Sciences 5 (4): 199-208. https://doi.org/10.33434/cams.1195074.

EndNote

Saleh M, Muraa I (December 1, 2022) On Weakly 1-Absorbing Primary Ideals of Commutative Semirings. Communications in Advanced Mathematical Sciences 5 4 199–208.

IEEE

[1]M. Saleh and I. Muraa, “On Weakly 1-Absorbing Primary Ideals of Commutative Semirings”, Communications in Advanced Mathematical Sciences, vol. 5, no. 4, pp. 199–208, Dec. 2022, doi: 10.33434/cams.1195074.

ISNAD

Saleh, Mohammad - Muraa, Ibaa. “On Weakly 1-Absorbing Primary Ideals of Commutative Semirings”. Communications in Advanced Mathematical Sciences 5/4 (December 1, 2022): 199-208. https://doi.org/10.33434/cams.1195074.

JAMA

1.Saleh M, Muraa I. On Weakly 1-Absorbing Primary Ideals of Commutative Semirings. Communications in Advanced Mathematical Sciences. 2022;5:199–208.

MLA

Saleh, Mohammad, and Ibaa Muraa. “On Weakly 1-Absorbing Primary Ideals of Commutative Semirings”. Communications in Advanced Mathematical Sciences, vol. 5, no. 4, Dec. 2022, pp. 199-08, doi:10.33434/cams.1195074.

Vancouver

1.Mohammad Saleh, Ibaa Muraa. On Weakly 1-Absorbing Primary Ideals of Commutative Semirings. Communications in Advanced Mathematical Sciences. 2022 Dec. 1;5(4):199-208. doi:10.33434/cams.1195074