We need to verify if this is a subring with or without identity of M(R). The book says it is without identity, but I am confused.

It is:

all 2x2 matrices of the form (0 r; 0 0) for r rational.

I don't see how this doesn't have identity since (0 r; 0 0)* (1 0; 0 1)=(0 r; 0 0)

Thanks.