Finding Prime Fields of the Ring Z_3 x Z_5

I have a ring R = Z_3 x Z_5 and I have to find its two prime fields.

I have tried to find out what these minimal rings could be. However, if the ring must have elements (0,0) and (1,1), then it would also have to have elements a - b according to the subring criterion, which would cause the ring to have (0,0) - (1,1) = (-1,-1) = (2,4). Now also (2,4) - (1,1) would have to be in subring S, so eventually this would make the whole ring R. This leads to the fact that R has no subrings, which makes me confused, since the task is to find two of them.

If I make the assumption that the ring does NOT need to have (1,1) however, then I could get two subrings (which are also prime fields, I think?):

S1 ={ (0,0), (0,1), (0,2), (0,3), (0,4) } and

S2 = { (0,0), (1,0), (2,0) }

Can I assume that the subrings do not have (1,1)?

Are S1 and S2 prime fields of Z_3 x Z_5 ?

If not, then some advice would be more than welcome.