Let $R$ be a non commutative prime ring without unity and let $S$ be the set of all ordered pair $(a,i)$, where $a\in R$ and $i\in I$, where $I$ is the ring of integers. On $S$ we define addition and multiplication as follows: $(a,i)+(b,j)=(a+b,i+j)$, $(a,i)(b,j)=(ab+ib+ja, ij)$. Then $S$ is a ring with unity $(0,1)$. My question- Is the ring $S$ a prime ring?
In case $R$ is a commutative prime ring with unity, I have shown $S$ is not a prime ring. But I am stuck in the case $R$ is non commutative prime ring without unity.