Is an ideal $A$ of a subsemigroup $H$ of a semigroup $G$ also an ideal of the semigroup $G$?

Question

Is an ideal $A$ of a subsemigroup $H$ of a subsemigroup $G$ also an ideal of the semigroup $G$?

Suppose we're talking about right ideals, specifically. We have that $Ah\in A$ for $h\in H$. We would need to show $Ag\in A$ for $g\in G$, if $A$ were to be a right ideal of $G$ also. I can't decide how to prove it because I can't decide what the case is likely to be.

I guess its probably not the case that $A$ is also an ideal of $G$. So if you suppose it is, you should find a contradiction (i.e., that $A$ is not an ideal of $H$ either) - but I'm not getting anywhere with this.

Can anyone help me with this (at least hint at which is actually is the case)?

Answer 1

No, that is not true. For example note that if $H$ is a subsemigroup of $G$ (i.e. $H^2\subset H$) then $H$ is trivially an ideal of $H$. But not every such $H$ is an ideal of $G$.

For example take $G$ the additive (semi)group $\mathbb Z[X]$, $H$ the sub(semi)group $\mathbb Z$. Any ideal $A$ (apart from $\{0\}$) of $\mathbb Z$ is not an ideal of $G$, as for example $A+X$ is not even in $H$.

Answer 2

Take a non-trivial monoid $G$. Then $H = \{1\}$ is a subsemigroup, $H$ is an ideal of $H$, but the right ideal of $G$ generated by $H$ is $G$. Thus the answer to your question is no.