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## Weakly prime ideals issued from an amalgamated algebra

#### Najib MAHDOU [1] , Moutu ABDOU SALAM MOUTUİ [2] , Youssef ZAHİR [3]

Let $R$ be a commutative ring with identity. A proper ideal $P$ is said to be weakly prime ideal of $R$ if for every $0\neq ab\in P$ where $a,b\in R,$ implies $a\in P$ or $b\in P$. The notion of weakly prime ideal was introduced by Anderson et al. in [Weakly prime ideals, Houston J. Math., 2003] as a generalization of prime ideals. The purpose of this paper is to study the form of weakly prime ideals of amalgamation of $A$ with $B$ along $J$ with respect to $f$ (denoted by $A\bowtie^{f}J$), introduced and studied by D'Anna et al. in [Amalgamated algebras along an ideal, Commutative Algebra and Its Applications, 2009]. Our results provide new techniques for the construction of new original examples of weakly prime ideals. Furthermore, as an application of our results, we provide an upper
bound for the weakly Krull dimension of amalgamation.
Amalgamated algebra, weakly prime ideal, weakly Krull dimension, amalgamated duplication
• [1] A.G. Agargun, D.D. Anderson and S. Valdes-Leon, Unique factorization ring with zero divisors, Comm. Algebra 27, 1967-1974, 1999.
• [2] K. Alaoui Ismaili and N. Mahdou, Coherence in amalgamated algebra along an ideal, Bull. Iranian Math. Soc. 41, 625-632, 2015.
• [3] D.D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math. 29 (4), 831-840, 2003.
• [4] D.D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra 1 (1), 3-56, 2009.
• [5] A. Badawi, On weakly semiprime ideals of commutative rings, Beitr. Algebra Geom. 51 (4), 1163-1173, 2014.
• [6] M. Boisen and P.B. Sheldon, CPI-extensions : overrings of integral domains with special prime spectrums, Canad. J. Math. 29, 722-737, 1977.
• [7] M. Chhiti, M. Jarrar, S. Kabbaj and N. Mahdou, Prüfer-like conditions in the amalgamated duplication of a ring along an ideal, Comm. Algebra 43 (1), 249-261, 2015.
• [8] M. D’Anna, C.A. Finocchiaro and M. Fontana, Amalgamated algebras along an ideal, Commutative Algebra and Its Applications, Walter de Gruyter, Berlin, 241-252, 2009.
• [9] M. D’Anna, C.A. Finocchiaro and M. Fontana, Properties of chains of prime ideals in an amalgamated algebra along an ideal, J. Pure Appl. Algebra, 1633-1641, 2010.
• [10] M. D’Anna and M. Fontana, The amalgamated duplication of ring along an ideal: the basic properties, J. Algebra Appl. 6 (3), 241-252, 2007.
• [11] S. Kabbaj, K. Louartiti and M. Tamekkante, Bi-amalgamated algebras along ideals, J. Commut. Algebra 9 (1), 65-87, 2017.
• [12] S. Kabbaj, N. Mahdou and M. A. S. Moutui, Bi-amalgamations subject to the arithmetical property, J. Algebra Appl. 16 (2), 11 pages, 2017.
• [13] N. Mahdou and M.A.S. Moutui, On (A)-rings and strong (A)-rings issued from amalgamations, Stud. Sci. Math. Hung. 55 (2), 270-279, 2018.
• [14] N. Mahdou and Y. Zahir, On weakly prime and weakly semiprime ideals, submitted.
Primary Language en Mathematics Mathematics Orcid: 0000-0001-6353-1114Author: Najib MAHDOU Institution: University S.M. Ben Abdellah FezCountry: Morocco Orcid: 0000-0002-7544-2749Author: Moutu ABDOU SALAM MOUTUİ (Primary Author)Institution: King Faisal UniversityCountry: Saudi Arabia Orcid: 0000-0002-5121-8822Author: Youssef ZAHİR Institution: University S.M. Ben Abdellah FezCountry: Morocco Publication Date : June 2, 2020
 Bibtex @research article { hujms557437, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1159 - 1167}, doi = {10.15672/hujms.557437}, title = {Weakly prime ideals issued from an amalgamated algebra}, key = {cite}, author = {Mahdou, Najib and Abdou Salam Moutui̇, Moutu and Zahi̇r, Youssef} } APA Mahdou, N , Abdou Salam Moutui̇, M , Zahi̇r, Y . (2020). Weakly prime ideals issued from an amalgamated algebra . Hacettepe Journal of Mathematics and Statistics , 49 (3) , 1159-1167 . DOI: 10.15672/hujms.557437 MLA Mahdou, N , Abdou Salam Moutui̇, M , Zahi̇r, Y . "Weakly prime ideals issued from an amalgamated algebra" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1159-1167 Chicago Mahdou, N , Abdou Salam Moutui̇, M , Zahi̇r, Y . "Weakly prime ideals issued from an amalgamated algebra". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1159-1167 RIS TY - JOUR T1 - Weakly prime ideals issued from an amalgamated algebra AU - Najib Mahdou , Moutu Abdou Salam Moutui̇ , Youssef Zahi̇r Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.557437 DO - 10.15672/hujms.557437 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1159 EP - 1167 VL - 49 IS - 3 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.557437 UR - https://doi.org/10.15672/hujms.557437 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Weakly prime ideals issued from an amalgamated algebra %A Najib Mahdou , Moutu Abdou Salam Moutui̇ , Youssef Zahi̇r %T Weakly prime ideals issued from an amalgamated algebra %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 3 %R doi: 10.15672/hujms.557437 %U 10.15672/hujms.557437 ISNAD Mahdou, Najib , Abdou Salam Moutui̇, Moutu , Zahi̇r, Youssef . "Weakly prime ideals issued from an amalgamated algebra". Hacettepe Journal of Mathematics and Statistics 49 / 3 (June 2020): 1159-1167 . https://doi.org/10.15672/hujms.557437 AMA Mahdou N , Abdou Salam Moutui̇ M , Zahi̇r Y . Weakly prime ideals issued from an amalgamated algebra. Hacettepe Journal of Mathematics and Statistics. 2020; 49(3): 1159-1167. Vancouver Mahdou N , Abdou Salam Moutui̇ M , Zahi̇r Y . Weakly prime ideals issued from an amalgamated algebra. Hacettepe Journal of Mathematics and Statistics. 2020; 49(3): 1159-1167.

Authors of the Article
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