The question is

Show that if I in M2(R) contains ( 1 0 ), then I=M2(R)

0 0

I know the fact that if I contains some unit then it would be equal to the ring R.

So,

( a b ) (1 0 ) = (a 0 )

c d 0 0 c 0

(1 0 ) ( A B ) = ( A B

0 0 C D 0 0

But how can i say that this is an ideal?