About the connection between ideals and homomorphisms

Question

I know that for a homomorphism of rings $\psi : R\rightarrow S$ we have that $\ker\psi$ is an ideal of $R$.

I was wondering if the opposite direction is true: Let $I$ be an ideal of $R$. Then does there exists a homomorphism $\psi : R\rightarrow S$ for some ring $S$ such that $\ker\psi = I$?

Answer 1

Yes! If $I$ is an ideal of $R$, then the projection map $$ \psi:R\to R/I $$ has $\ker \psi=I$.