Embedding of a ring into a ring with unity I was reading the theorem on Embedding of a ring into a ring with unity which is as follows:  

Let R be ring and $R\times \mathbb Z=\{(r,n)|r\in R,n\in \mathbb Z\}$. 
  This is a ring with addition defined as $(r,n)+(s,m)=(r+s,n+m).$ and multiplication defined as :$(r,n).(s,m)=(rs+ns+mr,nm)$ .this ring has unity as $(0,1)$.  
Now we can easily show that
  If we define an homomorphism from $R\to R_1$ as $f(r)=(r,0)$  $\forall r\in R$ then $R \cong f(R)\subseteq R_1$.
  Hence $R$ is embedable in $R_1$ which has unity $(0,1)$.

I can't understand why in the ring $R\times \mathbb Z=\{(r,n)|r\in R,n\in \mathbb Z\}$ we had to define multiplication as $(r,n).(s,m)=(rs+ns+mr,nm)$, why can't we define it as $(r,n).(s,m)=(rs,nm)$ ?

it may be helpful to look on the ring with unity as 
$$
eR \oplus \mathbb{Z}
$$
where $e$ is an idempotent which commutes with all other elements and satisfies  $e^2=e$
then every element is of the form $er +m$ and we have:
$$
(er+m)(es+n) = e^2rs+ern+mes +mn = e(rs+nr+ms) + mn
$$
the unity is the element $1$
