Remark

For the set S to be a subring it suffices to require that 0, 1, x - y and x * y are in S. Indeed, then 0 - x = -x is also in S.

A subring, with all restrictions of operations, is itself a ring.

The verification is almost trivial: Simply restrict the quantifiers to S; e.g., since

x * (y + z) = x * y + x * z

holds for all x, y, z R, it also holds for all elements in the subset S of R.