Exhibit the left and right ideals of a zero semigroup and a left zero semigroup

Find all of the left- and right ideals of a zero semigroup and of a left zero semigroup.


Let S be a zero semigroup. That is, there exists 0 \in S such that ab = 0 for all a,b \in S. In particular, S is commutative, so that left and right ideals coincide.

Let A \subseteq S be a subset containing 0. (In particular, A \neq \emptyset.) Then for all latex a \in A$ and s \in S, we have sa = 0 \in A. So SA \subseteq A, and thus A is an ideal.

Now let S be a left zero semigroup. That is, for all a,b \in S, we have ab = a.

Suppose A is a left ideal. Since A \neq \emptyset, there exists an element a \in A. Then for all s \in S, we have s = sa \in A. So in fact A = S, and S is the only left ideal. (I.e., S is left simple.)

Now let B \subseteq S be a nonempty subset with b \in B. For all s \in S, we have bs = b \in B. So BS \subseteq B, and thus B is a right ideal. That is, every nonempty subset of S is a right ideal.

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