Q: let R be a ring of 2 by 2 matrices with integer entries.which of the following subset of R is integral domain?
a) M(0,x,y,0)
b)M(x,0,0,y)
c)M(x,0,0,x)
d)M(x,y,y,z)
where M stands matrix of elements taken row by row.
Here you just need to apply the definitions of an integral domain.
Say that (in first case) $\displaystyle \begin{bmatrix}0&a\\b&0\end{bmatrix} \begin{bmatrix}0&c\\d&0\end{bmatrix} = \begin{bmatrix} 0&0\\0&0 \end{bmatrix}$ so $\displaystyle \begin{bmatrix} ad&0\\0&bc \end{bmatrix} = \begin{bmatrix} 0&0\\0&0 \end{bmatrix}$ which means $\displaystyle ad=0,bc=0$ and so $\displaystyle a=b=c=d=0$. So the only way to get the zero matrix is to have zero matrices in the very beginning. Try the same approach for the other three.