Proving that every ring $R$ admits a unique homomorphism $\mathbb{Z} \to R$

Question

I am trying to prove that there is a unique homomorphism between every ring, R and the integers, $\mathbb{Z}$.

I suggested that such a homomorphism $\phi : \mathbb{Z} \to R$ could be defined by for $ a \in \mathbb{Z}, \,\phi(a) \mapsto a \times 1_R$, where by '$\times$' I mean $1_R + 1_R + ...$ (summing $1_R$, a times).

Is this a suitable mapping?