Usual arithmetic
Modular arithmetic.
Polynomial rings.
Residue class rings
The Gaussian integers.
In the ring of integers Z, the intersection of the ideals
(m) and (n), for given integers m, n,
is the ideal (lcm(m,n)).
For, this is clear if at least one of m, n is zero. Otherwise,
if a 
m 
(n),
then a is a multiple of both m and n, and hence also of lcm(m,n).
Thus, m (n) is contained in (lcm(m,n)).
The other inclusion is obvious.
In the ring Z/nZ, the intersection of the ideals
(g) and (h) is (lcm(g,h)).
Just as for integers, the intersection of (f) and (g)
is (lcm(f,g)).
Just as for modular arithmetic, the intersection of (f) and (g)
is (lcm(f,g)).
In the ring R = Z+Zi, the intersection of
the ideals (1 + i) and (2) is (1 + i), as
2 = (1 - i)(1 + i).