Non-zero non-unitary ring homomorphisms [collecting examples]

Question

I am interested of finding examples of non-zero homomorphisms $f:R\to S$ of rings with unity such that $f(1_R)\neq 1_S$.

I will provide one example and I will be glad if others can also give examples.

Answer 1

Let be $f:\mathbb Z \to M_2(\mathbb Z); f(a)= \left( \array{a&0\\0&0}\right) $ then $f$ is ring homomorphism and $f(1)=\left(\array{1&0\\0&0}\right) \ne \left(\array{1&0\\0&1}\right)$ .

Answer 2

Let $R=\mathbb{Z}$ and $S=\mathbb{Z}\times \mathbb{Z}$. Define $f:R\to S$ by $f(n)=(n,0)$ for all $n\in\mathbb{Z}$. Then $f$ is a nonzero homomorphism but $$f(1_R)=f(1)=(1,0)\neq (1,1)=1_S.$$