Proving an ideal in a ring help

Is the set J= {0, 0; 0 r) | r exists in R (the reals) } an ideal in the ring M(R) of 2 x2 matrices over R.

It is not.
In order for a subset to be an ideal, multiplying an element of the ideal by any element of the ring must give you another element in the ideal.

Let matrix A be in J, and matrix M be in M(R).
WLOG,
M = (a,b; c,d)
A = (0,0; 0,r)
M*A = (0, b*r; 0, d*r)
Thus, J is not closed under multiplication in M(R)