Z is a subring of each. It is the smallest possible subring.

Q is a subring of R and of C.

R is a subring of C.

Let S be a subring. The S contains 1, hence each integer multiple of 1, hence the whole ring. Therefore, the only subring is the ring itself. In other words, there are no proper subrings.

The coefficient ring R is a subring.

Also the polynomials in which only even powers of X occur.

If f has positive degree, then, by the Lemma on page 3 of Section 4.2, R is a subring of R[X]/(f).

Z is a subring.

The upper triangular matrices form a subring.

Re is a subring. In fact e is the unit element, and so this ring is just a copy of R.

Also R + Ri is a subring, and similarly R + Rj and R + Rk .