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On divided and regular divided rings

Year 2019, Volume: 48 Issue: 5, 1454 - 1460, 08.10.2019

Abstract

In this paper, we study the notion of divided and regular divided rings. Then we establish the transfer of these notions to trivial ring extension and amalgamated algebras along an ideal. These results provide examples of non-divided regular divided rings. The article includes a brief discussion of the scope and precision of our results.

References

  • [1] K. Alaoui Ismaili and N. Mahdou, n-coherence property in amalgamated algebra along an ideal, Acta Math. Univ. Comenian. (N.S.), 86 (1), 59–72, 2017.
  • [2] A. Badawi, On divided commutative rings, Comm. Algebra, 27, 1465-1474, 1999.
  • [3] A. Badawi and D.E. Dobbs, On locally divided rings and going-down rings, Comm. Algebra, 29, 2805-2825, 2001.
  • [4] C. Bakkari, S. Kabbaj and N. Mahdou, Trivial extension defined by Prüfer condition, J. Pure Appl. Algebra, 214, 53–60, 2010.
  • [5] M. D’Anna and M. Fontana, The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat. 45 (2), 241–252, 2007.
  • [6] M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl. 6 (3), 443–459, 2007.
  • [7] M. D’Anna, C.A. Finocchiaro and M. Fontana, Amalgamated algebras along an ideal, in: Commutative algebra and its applications, Walter de Gruyter, Berlin, 241–252, 2009.
  • [8] M. D’Anna, C.A Finocchiaro and M. Fontana, Properties of prime ideals in an amalgamated algebra along an ideal, J. Pure Appl. Algebra, 214, 1633–1641, 2010.
  • [9] M. D’Anna, C.A Finocchiaro and M. Fontana, Algebraic and topological properties of an amalgamated algebra along an ideal, Comm. Algebra, 44, 1836-1851, 2016.
  • [10] D.E. Dobbs, Divided rings and going-down, Pac. J. Math. 67 (2), 353-363, 1976.
  • [11] D.E. Dobbs and J. Shapiro, A generalization of divided domains and its connection to weak Baer going-down rings, Comm. Algebra, 37, 3553-3572, 2009.
  • [12] S. Glaz, Commutative Coherent Rings, Springer-Verlag, Lecture Notes in Mathe- matics, 1371, 1989.
  • [13] J.A. Huckaba, Commutative Rings with Zero Divizors, Marcel Dekker, New York Basel, 1988.
  • [14] S. Kabbaj and N. Mahdou, Trivial Extensions Defined by coherent-like condition, Comm. Algebra, 32 (10), 3937–3953, 2004.
  • [15] N. Mahdou and M.A. Moutui, fqp-property in amalgamated algebras along an ideal, Asian-Eur. J. Math. 8 (3), 2015.
  • [16] N. Mahdou, M. Chhiti and M. Tamekkante Clean property in Amalgamated algebras along an ideal, Hacet. J. Math. Stat. 44 (1), 41–49, 2015.
  • [17] N. Mahdou, A. Mimouni and M.A. Moutui, On almost valuation and almost Bézout rings, Comm. Algebra, 43, 297–308, 2015.
  • [18] H.R. Maimani and S. Yassemi, Zero-divisor graphs of amalgamated duplication of a ring along an ideal, J. Pure Appl. Algebra, 212 (1), 168–174, 2008.
  • [19] G. Picavet, Ideals and overring of divided domains, Int. Electron. J. Algebra, 8, 80–113, 2009.
Year 2019, Volume: 48 Issue: 5, 1454 - 1460, 08.10.2019

Abstract

References

  • [1] K. Alaoui Ismaili and N. Mahdou, n-coherence property in amalgamated algebra along an ideal, Acta Math. Univ. Comenian. (N.S.), 86 (1), 59–72, 2017.
  • [2] A. Badawi, On divided commutative rings, Comm. Algebra, 27, 1465-1474, 1999.
  • [3] A. Badawi and D.E. Dobbs, On locally divided rings and going-down rings, Comm. Algebra, 29, 2805-2825, 2001.
  • [4] C. Bakkari, S. Kabbaj and N. Mahdou, Trivial extension defined by Prüfer condition, J. Pure Appl. Algebra, 214, 53–60, 2010.
  • [5] M. D’Anna and M. Fontana, The amalgamated duplication of a ring along a multiplicative-canonical ideal, Ark. Mat. 45 (2), 241–252, 2007.
  • [6] M. D’Anna and M. Fontana, An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra Appl. 6 (3), 443–459, 2007.
  • [7] M. D’Anna, C.A. Finocchiaro and M. Fontana, Amalgamated algebras along an ideal, in: Commutative algebra and its applications, Walter de Gruyter, Berlin, 241–252, 2009.
  • [8] M. D’Anna, C.A Finocchiaro and M. Fontana, Properties of prime ideals in an amalgamated algebra along an ideal, J. Pure Appl. Algebra, 214, 1633–1641, 2010.
  • [9] M. D’Anna, C.A Finocchiaro and M. Fontana, Algebraic and topological properties of an amalgamated algebra along an ideal, Comm. Algebra, 44, 1836-1851, 2016.
  • [10] D.E. Dobbs, Divided rings and going-down, Pac. J. Math. 67 (2), 353-363, 1976.
  • [11] D.E. Dobbs and J. Shapiro, A generalization of divided domains and its connection to weak Baer going-down rings, Comm. Algebra, 37, 3553-3572, 2009.
  • [12] S. Glaz, Commutative Coherent Rings, Springer-Verlag, Lecture Notes in Mathe- matics, 1371, 1989.
  • [13] J.A. Huckaba, Commutative Rings with Zero Divizors, Marcel Dekker, New York Basel, 1988.
  • [14] S. Kabbaj and N. Mahdou, Trivial Extensions Defined by coherent-like condition, Comm. Algebra, 32 (10), 3937–3953, 2004.
  • [15] N. Mahdou and M.A. Moutui, fqp-property in amalgamated algebras along an ideal, Asian-Eur. J. Math. 8 (3), 2015.
  • [16] N. Mahdou, M. Chhiti and M. Tamekkante Clean property in Amalgamated algebras along an ideal, Hacet. J. Math. Stat. 44 (1), 41–49, 2015.
  • [17] N. Mahdou, A. Mimouni and M.A. Moutui, On almost valuation and almost Bézout rings, Comm. Algebra, 43, 297–308, 2015.
  • [18] H.R. Maimani and S. Yassemi, Zero-divisor graphs of amalgamated duplication of a ring along an ideal, J. Pure Appl. Algebra, 212 (1), 168–174, 2008.
  • [19] G. Picavet, Ideals and overring of divided domains, Int. Electron. J. Algebra, 8, 80–113, 2009.
There are 19 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Chahrazade Bakkari This is me 0000-0002-6387-7972

Publication Date October 8, 2019
Published in Issue Year 2019 Volume: 48 Issue: 5

Cite

APA Bakkari, C. (2019). On divided and regular divided rings. Hacettepe Journal of Mathematics and Statistics, 48(5), 1454-1460.
AMA Bakkari C. On divided and regular divided rings. Hacettepe Journal of Mathematics and Statistics. October 2019;48(5):1454-1460.
Chicago Bakkari, Chahrazade. “On Divided and Regular Divided Rings”. Hacettepe Journal of Mathematics and Statistics 48, no. 5 (October 2019): 1454-60.
EndNote Bakkari C (October 1, 2019) On divided and regular divided rings. Hacettepe Journal of Mathematics and Statistics 48 5 1454–1460.
IEEE C. Bakkari, “On divided and regular divided rings”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, pp. 1454–1460, 2019.
ISNAD Bakkari, Chahrazade. “On Divided and Regular Divided Rings”. Hacettepe Journal of Mathematics and Statistics 48/5 (October 2019), 1454-1460.
JAMA Bakkari C. On divided and regular divided rings. Hacettepe Journal of Mathematics and Statistics. 2019;48:1454–1460.
MLA Bakkari, Chahrazade. “On Divided and Regular Divided Rings”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, 2019, pp. 1454-60.
Vancouver Bakkari C. On divided and regular divided rings. Hacettepe Journal of Mathematics and Statistics. 2019;48(5):1454-60.