Z is generated by {0,1} or, if you like, the empty set. For, 0,1 always belong to a subring; but then also -1 (because the additive structure is a group), 2 = 1+1, 3= 2+1, -2 = (-1)*2, ...

Q is not even finitely generated (that is, generated by a finite subset): to see this, use that there are infinitely many primes and study the possible denominators of element from a finitely generated subring.

Similarly, neither R nor C are finitely generated.

The ring Z/nZ is generated by the empty set.

The ring Z[X] is generated by X.

More generally, if R is a ring, its polynomial ring R[X] is generated by R {X}.

If R is a ring and f is a polynomial in R[X], then R[X]/(f) is generated by R {X}.

The ring R = Z+Zi of Gaussian integers is generated by i.

Let R be field. Then the matrix ring M_n(R) is generated by all upper and lower triangular matrices.

If R is a ring, the ring of quaternions over R:

H = R + Ri + Rj + Rk

with operations as in the example of Section 7.1, is generated by R, i, and j.