This is part $(c)$ and part $(e)$ of Exercise 1.9.5 of Howie's "Fundamentals of Semigroup Theory".
Definition: A non-empty subset $A$ of a semigroup $S$ is a right ideal if $AS\subseteq A$.
Let $\phi: S\to T$ be a morphism, where $S$ and $T$ are semigroups.
Show that if $\phi$ is onto then the image under $\phi$ of a right ideal of $S$ is a right ideal of $T$.
Show that the hypothesis that $\phi$ is onto cannot be removed.
Let $\emptyset\neq R\subseteq S$ be a right ideal of $S$. Then $RS\subseteq R.$ We have for all $r_1s\in RS$ there exists $r_2\in R$ such that $r_1s=r_2$.
Let $\phi: S\to T$ be onto. Then for all $t\in T$ there exists $\sigma\in S$ such that $\sigma\phi=t$.
We have $r_1\phi s\phi=r_2\phi$.
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