An example of a function failing to be a ring homomorphism

Question

Can anyone give an example of two rings $R$ and $S$ and a function $f$:$R$ → $S$ which preserves multiplication and addition but with $f$($1_R$) $\neq$ $1_S$. Thus $f$ failing to be a ring homomorphism.

Answer 1

Consider $R=\mathbb{Z}$ and $S=\mathbb{Z}\times\mathbb{Z}$ and the map $f\colon R\to S$ defined by $f(x)=(x,0)$.