# 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

## Access Full Article

top## Abstract

top## How to cite

topAnjan 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] 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] 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] 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] R.A. Good and D.R. Hughes, Associated groups for a semigroup, Bull. Amer. Math. Soc. 58 (1952), 624-625.
- [5] M. Henricksen, Ideals in semirings with commutative addition, Amer. Math. Soc. Notices (1958), 321.
- [6] A. Iampan, On bi-ideals of semigroups, Lobachevskii J. Math 29 (2) (2008), 68-72. doi: 10.1134/S1995080208020042 Zbl1165.20321
- [7] K.M. Kapp, On bi-ideals and quasi-ideals in semigroups, Publ. Math. Debrecan 16 (1969), 179-185. Zbl0224.20055
- [8] K.M. Kapp, Bi-ideals in associative rings and semigroups, Acta. Sci. Math. 33 (1972), 307-314. Zbl0247.20069
- [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] 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] S. Lajos, On (m, n)-ideals of semigroups, Abstract of Second Hunger. Math. Congress I (1960), 42-44.
- [12] S. Lajos, On the bi-ideals in semigroups, Proc. Japan. Acad. 45 (1969), 710-712. doi: 10.3792/pja/1195520625 Zbl0199.33702
- [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] S. Lajos and F. Szaaz, Bi-ideals in associative rings, Acta. Sci. Math. 32 (1971), 185-193. Zbl0217.34201
- [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] M. M. Miccoli, Bi-ideals in regular semigroups and orthogroups, Acta. Math. Hung. 47 (1-2) (1986), 3-6. doi: 10.1007/BF01949118
- [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] M.K. Sen and A.K. Bhuniya, Completely k-regular semirings, Bull. Cal. Math. Soc. 97 (2005), 455-466. Zbl1092.16028
- [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] M.K. Sen and A.K. Bhuniya, On semirings whose additive reduct is semilattice, Communicated. Zbl1248.16038
- [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

## Citations in EuDML Documents

top## NotesEmbed ?

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