Is this homomorphism in general surjective?

Question

Let $R$ be a commutative ring and $I$ an ideal of $R$. Pick a fix $0 \not= a \in I$ and consider the map $\phi: R \to I$ given by $r \mapsto ra$. Is this map surjective?

Answer 1

No! This map is not surjective in general. Example: $R=k[x, y], I =(x,y), a=x.$

Added: The image of the map $\phi: R \to I, r \mapsto ra$ is the ideal of $R$ generated by the element $a.$ So the map is surjective if and only if $I = \langle a \rangle.$