Free ideals and real ideals of the ring of frame maps from P(R) to a frame

Document Type : Research Paper

Authors

1 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Postal Code 9617976487, Sabzevar, Iran

2 Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran.

Abstract

Let FP(L) (FP∗(L)) be   the f-rings   of all (bounded) frame maps from P(R) to a frame L. FP∞(L) is  the family of all f∈FP(L) such that  ↑f(−1n,1n) is compact for any n∈N and the subring  FPK(L) is the family of all   f∈FP(L) such that coz(f) is compact. We  introduce  and study  the concept of   real ideals in FP(L) and FP∗(L). We  show  that every maximal ideal of FP∗(L) is   real, and also  we study the relation between the conditions ``L is compact" and ``every maximal ideal of FP(L) is real''. We prove  that for every   nonzero real Riesz map φ:FP(L)→R,  there is an element  p in ΣL such that φ=pcoz~
  if L is a zero-dimensional frame for which B(L) is a sub-σ-frame  of   L and every maximal ideal of FP(L) is real. We show  that FP∞(L)  is equal to the intersection of all  free maximal ideals of FP∗(L) if B(L) is a sub-σ-frame  of a zero-dimensional frame  L   and also,  FPK(L) is equal to the intersection of all free ideals FP(L)   (resp.,  FP∗(L)) if L is a zero-dimensional frame.  Also, we study free ideals and fixed ideals of    FP∞(L) and  FPK(L).

Keywords

References

[1] S. K. Acharyya, G. Bhunia, and P.P. Ghosh, Finite frames, P-frames and basically disconnected frames, Algebra Univers. 72(3) (2014), 209{224. https://doi.org/10.1007/s00012--014--0296--x.
[2] S. Afrooz and M. Namdari, C∞(X) and related ideals, Real Anal. Exch. 36(1) (2010/2011), 45-54.
[3] A.R. Aliabad, F. Azarpanah, and M. Namdari, Rings of continuous functions vanishing at in nity, Comment. Math. Univ. Carolin. 45 (2004), 519-533.
[4] F. Azarpanah, Essential ideals in C(X), Period. Math. Hungar. 31(2) (1995), 105-112.
[5] F. Azarpanah and R. Soundararajan, When the family of functions vanishing at in nity is an ideal of C(X), Rocky Mt. J. Math. 31 (2001), 1133-1140. https://doi.org/10.1216/rmjm/1021249434.
[6] B. Banaschewski, The real numbers in pointfree topology, Textos de Matematica (Series B) 12 (1997), 1-96.
[7] B. Banaschewski, Remarks concerning certain function rings in pointfree topology, Appl. Categor. Struct. 26 (2018), 873-881. https://doi.org/10.1007/s10485--018--9514--6.
[8] A. Bigard, K. Keimel, and S. Wolfenstein, Groups et anneaux reticules, Series: Lecture Notes in Mathematics Vol. 608, Springer-Verlag, Berlin Heidelberg, 1997.
[9] T. Dube, Concerning P-frames, essential P-frames, and strongly zero-dimensional frames, Algebra Univers. 61 (2009), 115-138. https://doi.org/10.1007/s00012--009--0006--2.
[10] T. Dube, On the ideal of functions with compact support in pointfree function rings, Acta Math. Hungar 129(3) (2010), 205-226. https://doi.org/10.1007/s10474--010--0024--8.
[11] M.M. Ebrahimi, A. Karimi Feizabadi and M. Mahmoudi, Pointfree spectra of riesz maps, Appl. Categ. Struct. 12(4) (2004), 379-409. https://doi.org/10.1023/B:APCS.0000040553.07828.59.
[12] A.A. Estaji, M. Abedi, and A. Mahmoudi Darghadam, On self-injectivity of the f-ring Frm(P(R), L), Math. Slovaca 69(5) (2019), 999-1008. https://doi.org/10.1515/ms--2017--0284.
[13] A.As. Estaji, E. Hashemi, and A.A. Estaji, On property (A) and socle of real-valude functions on a frame, Categ. Gen. Algebr. Struct. Appl. 8(1) (2018), 61-80.
[14] A.A. Estaji, A. Karimi Feizabadi, and B. Emamverdi, Representation of real riesz maps on a strong f-ring by prime elements of a frame, Algebra Univers. 79 (2018), Pages14. https://doi.org/10.1007/s00012--018--0503--2.
[15] A.A. Estaji and A. Mahmoudi Darghadam, Rings of continuous functions vanishing at in nity on a frame,Quaest. Math. 42(9) (2019), 1141-1157. http://dx.doi.org/10.2989/16073606.2018.1509151.
[16] L. Gillman and M. Jerison, Rings of continuous functions, Springer-Verlag, 1976.
[17] J. Gutierrez Garca, J. Picado, and A. Pultr, Notes on point-free real functions and sublocales, Categorical methods in algebra and topology, 167-200, Textos de Matematica, DMUC 46, Univ. Coimbra (2014).
[18] P.T. Johnstone, Stone spaces, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1982.
[19] A. Karimi Feizabadi, A.A. Estaji, and B. Emamverdi, RL-valued f-ring homomorphisms and lattice-valued maps, Categ. Gen. Algebr. Struct. Appl. 8 (2017), 141-163.
[20] A. Karimi Feizabadi, A.A. Estaji, and M. Zarghani, The ring of real-valued functions on a frame, Categ. Gen. Algebr. Struct. Appl. 5(1) (2016), 85-102.
[21] A. Karimi Feizabadi and M.M. Ebrahimi, Pointfree prime representation of real Riesz maps, Algebra Univers. 54(3) (2005), 291-299. https://doi.org/10.1007/s00012--005--1945--x.
[22] C.W. Kohls, The space of prime ideals of a ring, Fund. Math. 45 (1957), 17-27.
[23] J. Picado and A. Pultr, Frames and locales: Topology without points, Frontiers in Mathematics, Springer Basel, 2012.
[24] D. Rudd, On isomorphisms between ideals in rings of continuous functions, Trans. Amer. Math. Soc 159 (1971), 335-353. https://doi.org/10.2307/1996015.
[25] M. Zarghani and A. Karimi Feizabadi, Zero elements in lattice theory, Extended Abstracts of the 25th Iranian Algebra Seminar, Hakim Sabzevari University, Sabzevar, Iran (July 20-21, 2016).
Algebraic Structures and Their Applications
Volume 7, Issue 2
November 2020
Pages 93-113

Files

Share

How to cite

Statistics