Give an example of a 0-simple semigroup.
Recall that a semigroup with zero is called 0-simple if and has no ideals other than 0 and .
Let be a group, and consider the semigroup . (Recall that if is a semigroup, is the semigroup obtained by attaching a new element to which acts like a zero.) So is a semigroup with zero. Certainly . Now let be an ideal. If , then there exists an element such that . Now (since is a group). So , and certianly . So . Thus is 0-simple.