Originally Posted by

**spoon737** I'm trying to understand a particular proof, but it has a step which isn't really given any justification. Here is the part that's giving me trouble:

"Let $\displaystyle I$ be a semiprime ideal of a semigroup $\displaystyle S$, and let $\displaystyle d \in S-I$. Letting $\displaystyle M = \{d^n| n = 1, 2, ...\}$, we obtain an m-system disjoint from $\displaystyle I$."

For those unfamiliar with the terminology, a semiprime ideal is an ideal of a semigroup $\displaystyle S$ with the property that for any $\displaystyle a \in S, aSa \subseteq I \Longrightarrow a \in I$. An m-system is a non-empty subset $\displaystyle A$ of S such that $\displaystyle a,b \in A \Longrightarrow$ there exists $\displaystyle x \in S$ such that $\displaystyle axb \in S$.

What I'm trying to understand is why $\displaystyle M$ should be disjoint from $\displaystyle I$. I understand why it is an m-system, but I don't understand why, say, $\displaystyle d^2$ can't be in $\displaystyle I$. Any insights would be appreciated.