Öz
Kaynakça
- [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.
Öz
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.
Anahtar Kelimeler
Amalgamated algebra weakly prime ideal weakly Krull dimension amalgamated duplication
Kaynakça
- [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.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematik |
| Bölüm | Matematik |
| Yazarlar | |
| Yayımlanma Tarihi | 2 Haziran 2020 |
| Yayımlandığı Sayı | Yıl 2020 Cilt: 49 Sayı: 3 |