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Year 2020, Volume: 49 Issue: 1, 254 - 272, 06.02.2020
https://doi.org/10.15672/hujms.455030

Abstract

References

  • [1] A.R. Aliabad and M. Badie, Fixed-place ideals in commutative rings, Comment. Math. Univ. Carolin. 54 (1), 53–68, 2013.
  • [2] A.R. Aliabad and M. Badie, On Bourbaki associated prime divisors of an ideal, Quaest. Math. 42 (4), 479-500, 2019.
  • [3] A.R. Aliabad and R. Mohamadian, On $sz^{\circ}$-ideals in polynomial rings, Comm. Algebra 39 (2) (2011), 701–717, 2011.
  • [4] A.R. Aliabad, R. Mohamadian, and S. Nazari, On regular ideals in reduced rings, Filomat 31 (12), 3715–3726, 2017.
  • [5] A.R. Aliabad and S. Nazari, On the spectrum of a commutative ring via C(X), (to appear).
  • [6] A.R. Aliabad, A. Taherifar, and N. Tayarzadeh, $\alpha$-Baer rings and some related concepts via C(X), Quaest. Math. 39 (3), 401–419, 2016.
  • [7] G. Artico, U. Marconi, and R. Moresco, A subspace of Spec(A) and its connexions with the maximal ring of quotients, Rend. Sem. Mat. Univ., Padova 64, 93–107, 1981.
  • [8] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, vol. 2, Addison-Wesley Reading, 1969.
  • [9] F. Azarpanah, O.A.S. Karamzadeh, and A.R. Aliabad, On $z^{\circ}$-ideals in C(X), Fund. Math. 160 (1), 15-25,1999.
  • [10] F. Azarpanah, O.A.S. Karamzadeh, and A.R. Aliabad, On ideals consisting entirely of zero-divisors, Comm. Algebra 28 (2), 1061–1073, 2000.
  • [11] T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer-Verlag, London, 2005.
  • [12] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand Reinhold, New York, 1960.
  • [13] G. Grätzer, General Lattice Theory, Birkhäuser, Basel, 1998.
  • [14] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115, 110–130, 1965.
  • [15] C.B. Huijsmans and B. De Patger, On z-ideals and d-ideals in Riesz spaces. I, Indag. Math. (Proceeding) A 83 (2), 183–195, 1980.
  • [16] D.G. Johnson and M. Mandelker, Functions with pseudocompact support, Topology Appl. 3 (4), 331–338, 1973.
  • [17] J. Kist, Minimal prime ideals in commutative semigroups, Proc. Lond. Math. Soc. 3 (1), 31–50, 1963.
  • [18] G. Mason, z-ideals and prime ideals, J. Algebra 26 (2), 280–297, 1973.
  • [19] G. Mason, Prime ideals and quotient rings of reduced rings, Math. Jpn. 34 (6), 941– 956, 1989.
  • [20] Y. Quentel, Sur la compacité du spectre minimal d’un anneau, Bull. Soc. Math. France 99, 265–272, 1971.
  • [21] R.Y. Sharp, Steps in Commutative Algebra, Cambridge university press, London, 1990.
  • [22] S. Willard, General Topology, Addison Wesley, Reading Mass., New York, 1970.

An extension of $z$-ideals and $z^\circ$-ideals

Year 2020, Volume: 49 Issue: 1, 254 - 272, 06.02.2020
https://doi.org/10.15672/hujms.455030

Abstract

Let $R$ be a commutative ring, $Y\subseteq Spec(R)$ and $ h_Y(S)=\{P\in Y:S\subseteq P \}$, for every $S\subseteq R$. An ideal $I$ is said to be an $\mathcal{H}_Y$-ideal whenever it follows from $h_Y(a)\subseteq h_Y(b)$ and $a\in I$ that $b\in I$. A strong  $\mathcal{H}_Y$-ideal is defined in the same way by replacing an arbitrary finite set $F$ instead of the element $a$. In this paper these two classes of ideals (which are based on the spectrum of the ring $R$ and are a generalization of the well-known concepts semiprime ideal, z-ideal, $z^{\circ}$-ideal (d-ideal), sz-ideal and $sz^{\circ}$-ideal ($\xi$-ideal)) are studied. We show that the most important results about these concepts, Zariski topology", annihilator" and etc can be extended in such a way that the corresponding consequences seems to be trivial and useless. This comprehensive look helps to recognize the resemblances and differences of known concepts better.

References

  • [1] A.R. Aliabad and M. Badie, Fixed-place ideals in commutative rings, Comment. Math. Univ. Carolin. 54 (1), 53–68, 2013.
  • [2] A.R. Aliabad and M. Badie, On Bourbaki associated prime divisors of an ideal, Quaest. Math. 42 (4), 479-500, 2019.
  • [3] A.R. Aliabad and R. Mohamadian, On $sz^{\circ}$-ideals in polynomial rings, Comm. Algebra 39 (2) (2011), 701–717, 2011.
  • [4] A.R. Aliabad, R. Mohamadian, and S. Nazari, On regular ideals in reduced rings, Filomat 31 (12), 3715–3726, 2017.
  • [5] A.R. Aliabad and S. Nazari, On the spectrum of a commutative ring via C(X), (to appear).
  • [6] A.R. Aliabad, A. Taherifar, and N. Tayarzadeh, $\alpha$-Baer rings and some related concepts via C(X), Quaest. Math. 39 (3), 401–419, 2016.
  • [7] G. Artico, U. Marconi, and R. Moresco, A subspace of Spec(A) and its connexions with the maximal ring of quotients, Rend. Sem. Mat. Univ., Padova 64, 93–107, 1981.
  • [8] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, vol. 2, Addison-Wesley Reading, 1969.
  • [9] F. Azarpanah, O.A.S. Karamzadeh, and A.R. Aliabad, On $z^{\circ}$-ideals in C(X), Fund. Math. 160 (1), 15-25,1999.
  • [10] F. Azarpanah, O.A.S. Karamzadeh, and A.R. Aliabad, On ideals consisting entirely of zero-divisors, Comm. Algebra 28 (2), 1061–1073, 2000.
  • [11] T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer-Verlag, London, 2005.
  • [12] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand Reinhold, New York, 1960.
  • [13] G. Grätzer, General Lattice Theory, Birkhäuser, Basel, 1998.
  • [14] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115, 110–130, 1965.
  • [15] C.B. Huijsmans and B. De Patger, On z-ideals and d-ideals in Riesz spaces. I, Indag. Math. (Proceeding) A 83 (2), 183–195, 1980.
  • [16] D.G. Johnson and M. Mandelker, Functions with pseudocompact support, Topology Appl. 3 (4), 331–338, 1973.
  • [17] J. Kist, Minimal prime ideals in commutative semigroups, Proc. Lond. Math. Soc. 3 (1), 31–50, 1963.
  • [18] G. Mason, z-ideals and prime ideals, J. Algebra 26 (2), 280–297, 1973.
  • [19] G. Mason, Prime ideals and quotient rings of reduced rings, Math. Jpn. 34 (6), 941– 956, 1989.
  • [20] Y. Quentel, Sur la compacité du spectre minimal d’un anneau, Bull. Soc. Math. France 99, 265–272, 1971.
  • [21] R.Y. Sharp, Steps in Commutative Algebra, Cambridge university press, London, 1990.
  • [22] S. Willard, General Topology, Addison Wesley, Reading Mass., New York, 1970.
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Ali Rezaei Aliabad This is me 0000-0003-1293-3652

Mehdi Badie 0000-0003-1114-3130

Sajad Nazari This is me 0000-0002-4295-2435

Publication Date February 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 1

Cite

APA Aliabad, A. R., Badie, M., & Nazari, S. (2020). An extension of $z$-ideals and $z^\circ$-ideals. Hacettepe Journal of Mathematics and Statistics, 49(1), 254-272. https://doi.org/10.15672/hujms.455030
AMA Aliabad AR, Badie M, Nazari S. An extension of $z$-ideals and $z^\circ$-ideals. Hacettepe Journal of Mathematics and Statistics. February 2020;49(1):254-272. doi:10.15672/hujms.455030
Chicago Aliabad, Ali Rezaei, Mehdi Badie, and Sajad Nazari. “An Extension of $z$-Ideals and $z^\circ$-Ideals”. Hacettepe Journal of Mathematics and Statistics 49, no. 1 (February 2020): 254-72. https://doi.org/10.15672/hujms.455030.
EndNote Aliabad AR, Badie M, Nazari S (February 1, 2020) An extension of $z$-ideals and $z^\circ$-ideals. Hacettepe Journal of Mathematics and Statistics 49 1 254–272.
IEEE A. R. Aliabad, M. Badie, and S. Nazari, “An extension of $z$-ideals and $z^\circ$-ideals”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 1, pp. 254–272, 2020, doi: 10.15672/hujms.455030.
ISNAD Aliabad, Ali Rezaei et al. “An Extension of $z$-Ideals and $z^\circ$-Ideals”. Hacettepe Journal of Mathematics and Statistics 49/1 (February 2020), 254-272. https://doi.org/10.15672/hujms.455030.
JAMA Aliabad AR, Badie M, Nazari S. An extension of $z$-ideals and $z^\circ$-ideals. Hacettepe Journal of Mathematics and Statistics. 2020;49:254–272.
MLA Aliabad, Ali Rezaei et al. “An Extension of $z$-Ideals and $z^\circ$-Ideals”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 1, 2020, pp. 254-72, doi:10.15672/hujms.455030.
Vancouver Aliabad AR, Badie M, Nazari S. An extension of $z$-ideals and $z^\circ$-ideals. Hacettepe Journal of Mathematics and Statistics. 2020;49(1):254-72.