Let R and R' be rings and let N and N' be ideals of R and R' respectively.

Let $\displaystyle \phi $ be a homomorphism of R into R'.

Show that $\displaystyle \phi $ induces a natural homomorphism $\displaystyle \phi_* : R/N \rightarrow R'/N' $ if $\displaystyle \ \ \phi [N] \subseteq N' $