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Let be a semigroup. An ideal of is a non-empty subset of which is closed under multiplication on either side by elements of . Formally, is an ideal of if is non-empty, and for all and , we have and .
One-sided ideals are defined similarly. A non-empty subset of is a left ideal (resp. right ideal) of if for all and , we have (resp. ).
A principal left ideal of is a left ideal generated by a single element. If , then the principal left ideal of generated by is
. (The notation is explained here.)
Similarly, the principal right ideal generated by is
.
The notation and are also common for the principal left and right ideals generated by respectively.
A principal ideal of is an ideal generated by a single element. The ideal generated by is
The notation
is also common.
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