 This remark holds more generally in any semigroup with zero.Furthermore, if every element has a unique pseudoinverse, then the semigroup is a group, and the unique pseudoinverse of an element coincides with the group inverse.Recall that the principal ideals of a semigroup "S" are defined in terms of "S"1, the "semigroup with identity adjoined"; this is to ensure that an element "a" belongs to the principal right, left and two-sided ideals which it generates.