For a semigroup (or ordered semigroup) $S$, we denote by

$\cal I$$(M)$ the ideal of $S$ generated by $M\;\,(M\subseteq S)$. 

In this note we prove the following: If $S$ is an ordered semigroup 

(or a semigroup), then a proper ideal $M$ of $S$ is a maximal ideal 

of $S$ if and only if for every $a\in S\backslash M$, we have 

$\cal I$$(M\cup \{a\}) = S$. If $S$ is a finitely generated ordered 

semigroup (or a semigroup), then each proper ideal of $S$ is 

contained in a maximal ideal of $S$. If $S$ is an ordered semigroup 

(or a semigroup) for which there exists an element $a$ of $S$ such 

that $\cal I$$(a)=S$, then each proper ideal of $S$ is contained 

in a maximal ideal of $S$. Similar results hold if, in the results 

above, we replace the word "ideal" by "left ideal" or "right ideal".