In the ring of integers Z, the subset (n) of all multiples of n forms an ideal: if an and bn are multiples of n, then an + bn = (a + b)n is a multiple of n; if furthermore r is in Z and an is a multiple of n then r(an) = (ra)n is a multiple of n.

In the ring Z/nZ, where n is a multiple of m Z, the set of all multiples of m is an ideal, denoted again by (m) or mZ/nZ.

In the polynomial ring R[X] the multiples of a given polynomial f form an ideal.

In Z[X] the subset {f | f(2) = 0} is an ideal:

• if f(2) = 0 and g(2) = 0 then (f + g)(2) = f(2) + g(2) = 0 + 0 = 0, and
• if f(2) = 0 and r is an element of Z[X], then (rf)(2) = r(2)f(2) = 0.

In R[X]/(f), the set of all multiples of the residue class of a divisor g of f is an ideal, denoted by (g) or gR[X]/(f).

In the ring R = Z + Zi, the set of all elements a + bi with a = b mod 2 is an ideal.