Bi-ideals in k-regular and intra k-regular semirings

Anjan K. Bhuniya; Kanchan Jana

Discussiones Mathematicae - General Algebra and Applications (2011)

  • Volume: 31, Issue: 1, page 5-23
  • ISSN: 1509-9415

Abstract

top
Here we introduce the k-bi-ideals in semirings and the intra k-regular semirings. An intra k-regular semiring S is a semiring whose additive reduct is a semilattice and for each a ∈ S there exists x ∈ S such that a + xa²x = xa²x. Also it is a semiring in which every k-ideal is semiprime. Our aim in this article is to characterize both the k-regular semirings and intra k-regular semirings using of k-bi-ideals.

How to cite

top

Anjan K. Bhuniya, and Kanchan Jana. "Bi-ideals in k-regular and intra k-regular semirings." Discussiones Mathematicae - General Algebra and Applications 31.1 (2011): 5-23. <http://eudml.org/doc/276570>.

@article{AnjanK2011,
abstract = {Here we introduce the k-bi-ideals in semirings and the intra k-regular semirings. An intra k-regular semiring S is a semiring whose additive reduct is a semilattice and for each a ∈ S there exists x ∈ S such that a + xa²x = xa²x. Also it is a semiring in which every k-ideal is semiprime. Our aim in this article is to characterize both the k-regular semirings and intra k-regular semirings using of k-bi-ideals.},
author = {Anjan K. Bhuniya, Kanchan Jana},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {k-bi-ideals; k-ideals; semiprimary subsets; k-regular semirings; intra k-regular semirings; quasi -ideals; -regular semirings; intra regular semirings; -bi-ideals},
language = {eng},
number = {1},
pages = {5-23},
title = {Bi-ideals in k-regular and intra k-regular semirings},
url = {http://eudml.org/doc/276570},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Anjan K. Bhuniya
AU - Kanchan Jana
TI - Bi-ideals in k-regular and intra k-regular semirings
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2011
VL - 31
IS - 1
SP - 5
EP - 23
AB - Here we introduce the k-bi-ideals in semirings and the intra k-regular semirings. An intra k-regular semiring S is a semiring whose additive reduct is a semilattice and for each a ∈ S there exists x ∈ S such that a + xa²x = xa²x. Also it is a semiring in which every k-ideal is semiprime. Our aim in this article is to characterize both the k-regular semirings and intra k-regular semirings using of k-bi-ideals.
LA - eng
KW - k-bi-ideals; k-ideals; semiprimary subsets; k-regular semirings; intra k-regular semirings; quasi -ideals; -regular semirings; intra regular semirings; -bi-ideals
UR - http://eudml.org/doc/276570
ER -

References

top
  1. [1] M.R. Adhikari, M.K. Sen and H.J. Weinert, On k-regular semirings, Bull.Cal.Math.Soc. 88 (1996), 141-144. Zbl0886.16032
  2. [2] S. Bourne, The Jacobson radical of a semiring, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 163-170. doi: 10.1073/pnas.37.3.163 Zbl0042.03201
  3. [3] R. Chinram and K. Tinpun, A note on minimal bi-ideals in ordered Γ-semigroups, International Math. Forum 4 (1) (2009), 1-5. Zbl1179.06006
  4. [4] R.A. Good and D.R. Hughes, Associated groups for a semigroup, Bull. Amer. Math. Soc. 58 (1952), 624-625. 
  5. [5] M. Henricksen, Ideals in semirings with commutative addition, Amer. Math. Soc. Notices (1958), 321. 
  6. [6] A. Iampan, On bi-ideals of semigroups, Lobachevskii J. Math 29 (2) (2008), 68-72. doi: 10.1134/S1995080208020042 Zbl1165.20321
  7. [7] K.M. Kapp, On bi-ideals and quasi-ideals in semigroups, Publ. Math. Debrecan 16 (1969), 179-185. Zbl0224.20055
  8. [8] K.M. Kapp, Bi-ideals in associative rings and semigroups, Acta. Sci. Math. 33 (1972), 307-314. Zbl0247.20069
  9. [9] N. Kehayopulu, J.S. Ponizovskii and M. Tsingelis, Bi-ideals in ordered semigroups and ordered groups, J. Math. Sci. 112 (4) (2002), 4353-4354. doi: 10.1023/A:1020347003781 
  10. [10] Y. Kemprasit, Quasi-ideals and bi-ideals in semigroups and rings, Proceedings of the International Conference on Algebra and its Applicatipns (2002), 30-46. Zbl1071.20515
  11. [11] S. Lajos, On (m, n)-ideals of semigroups, Abstract of Second Hunger. Math. Congress I (1960), 42-44. 
  12. [12] S. Lajos, On the bi-ideals in semigroups, Proc. Japan. Acad. 45 (1969), 710-712. doi: 10.3792/pja/1195520625 Zbl0199.33702
  13. [13] S. Lajos, On generalized bi-ideals in semigroups, Coll. Math. Soc. Janos Bolyai, Algebraic Theory of semigroups (G. Polak, Ed.), North-Holland 20 (1979), 335-340. 
  14. [14] S. Lajos and F. Szaaz, Bi-ideals in associative rings, Acta. Sci. Math. 32 (1971), 185-193. Zbl0217.34201
  15. [15] S. Li and Y. He, On semigroups whose bi-ideals are strongly prime, International J. Algebra 1 (6) (2007), 263-268. Zbl1127.20039
  16. [16] M. M. Miccoli, Bi-ideals in regular semigroups and orthogroups, Acta. Math. Hung. 47 (1-2) (1986), 3-6. doi: 10.1007/BF01949118 
  17. [17] J.V. Neumann, On regular rings, Proc. Nat. Acad. Sci. USA 22 (1936), 707-713. doi: 10.1073/pnas.22.12.707 Zbl0015.38802
  18. [18] M.K. Sen and A.K. Bhuniya, Completely k-regular semirings, Bull. Cal. Math. Soc. 97 (2005), 455-466. Zbl1092.16028
  19. [19] M.K. Sen and A.K. Bhuniya, On Additive Idempotent k-Clifford Semirings, Southeast Asian Bulletin of Mathematics 32 (2008), 1149-1159. Zbl1199.16103
  20. [20] M.K. Sen and A.K. Bhuniya, On semirings whose additive reduct is semilattice, Communicated. Zbl1248.16038
  21. [21] X.Z. Xu and J.Y. Ma, A note on minimal bi-ideals in ordered semigroups, SEA Bull. Math. 27 (2003), 149-154. Zbl1053.06011

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.