Embedding a Ring If I am looking at the following problem:

Let R be ring and $S=R\times \mathbb Z=\{(r,n)|r\in R,n\in \mathbb Z\}$. 
  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 $(0,1)$.  

How would I show that if $(R,R_{+},R_{*})$ has zero divisors then this will imply $(S,S_{+},S_{*})$   ? 
I saw similar problems but nothing showing this directly. 

Suppse that $rs=0$, $(r,0)(s,0)=(0,0)$ so $S$ has divisors of zero if $R$ has divisors of zero.
