IJPAM: Volume 83, No. 1 (2013)
Department of Mathematics
COMSATS Institute of Information Technology
Institute of Mathematics
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 -groupoid ) with left identity. It is proved that if is a minimal left ideal of an -groupoid with left identity then is a minimal left ideal of for all We also show that the kernel of an -groupoid ( the intersection of all two sided ideals of if exists)is simple and the class sum of all minimal left ideals of containing at least one minimal left ideal of is precisly the kernel of Finally, we show that if is an -groupoid with left identity then for all Finally, if is an -groupoid with left identity and does not contain any non-trivial nilpotent ideals, then every minimal ideal of 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 -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