In the ring of integers Z, the sum of the ideals (m) and (n), for given integers m, n 0, is the ideal (gcd(m,n)).

To see this, let a and b be integers such that am + bn = gcd(m,n) (they can be found by menas of the extended Euclidean algorithm). This equality shows that gcd(m,n), and therefore every multiple of it, belongs to the ideal generated by m and n. This shows that the ideal (gcd(m,n)) is contained in the ideal (m) + (n).

On the other hand, every element cm + dn of the sum ideal (m) + (n) is a multiple of gcd(m,n), since both m and n are multiples of this gcd.

Suppose m, n 0. If m and n divide d, then the sum of the ideals (m) and (n) of Z/dZ is the ideal (gcd(m,n)).

Just as for integers, in the polynomial ring Q[X], the sum of the ideals (f) and (g) equals the ideal (gcd(f,g)) whenever f, g 0.

Just as for modular arithmetic, the sum of (f) and (g) is (gcd(f,g)) whenever f, g 0.

In the ring R = Z + Zi, the sum of the ideals (1 + i) and (1 - i) is (1 + i), as 1 - i = -i(1 + i).