 If all rings have been assumed, by convention, to have a multiplicative identity, then to be a subring one would also require "S" to share the same identity element as "R".So if all rings have been assumed to have a multiplicative identity, then a proper ideal is not a subring.For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z["X"] (in both cases, Z contains 1, which is the multiplicative identity of the larger rings).