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 ∈ 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 sumof 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 ∈ 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 (3) concerning the minimal one sided ideals of semigroups and rings are consequently extended to and strengthened in AG-groupoids.
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