IJPAM: Volume 83, No. 1 (2013)




COMSATS Institute of Information Technology
Abbottabad, PAKISTAN

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 -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
Year: 2013
Volume: 83
Issue: 1