IJPAM: Volume 83, No. 1 (2013)

MINIMAL IDEALS OF ABEL-GRASSMANN GROUPOIDS

Madad Khan$^1$, K.P. Shum$^2$, M. Faisal Iqbal$^3$
$^{1,3}$Department of Mathematics
COMSATS Institute of Information Technology
Abbottabad, PAKISTAN
$^2$Institute of Mathematics
Yunan University
Kunming, 650091, P.R. CHINA


Abstract. The properties of Abel-Grassmann groupoids have been attracted the attention of many authors. The aim of this paper is to study the properties of the minimal left ideals of an Abel-Grassmann groupoid ( in brevity, an $AG$-groupoid ) with left identity. It is proved that if $L$ is a minimal left ideal of an $AG$-groupoid $S$ with left identity then $Lc$ is a minimal left ideal of $S$ for all $ c \in S.$ We also show that the kernel $K$ of an $AG$-groupoid $S$ ( the intersection of all two sided ideals of $S$ if exists)is simple and the class sum $\Sigma $ of all minimal left ideals of $S$ containing at least one minimal left ideal of $S$ is precisly the kernel $K$ of $S.$ Finally, we show that if $S$ is an $AG$-groupoid with left identity then $Sa^{2}S=Sa^{2}$ for all $a\in S.$ Finally, if $S$ is an $AG$-groupoid with left identity and does not contain any non-trivial nilpotent ideals, then every minimal ideal of $S$ is simple.A number of classical results of L. M. Gluskin and O. Steinfeld given in 1978 [#!__11__G-Steinfeld__11__!#] concerning the minimal one sided ideals of semigroups and rings are consequently extended to and strengthened in $AG$-groupoids.

Received: November 29, 2012

AMS Subject Classification: 20M10, 20N99

Key Words and Phrases: AG-groupoids, medial law, minimal ideals, 0-minimal ideals, nilpotent ideals

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DOI: 10.12732/ijpam.v83i1.11 How to cite this paper?
Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2013
Volume: 83
Issue: 1