Not every ring is a unique factorization domain. For instance, in the ring of numbers formula_193 (for integers formula_37 and formula_42) the number formula_196 has two factorizations formula_197, where neither of the four factors can be reduced any further, so it does not have a unique factorization. In order to extend unique factorization to a larger class of rings, the notion of a number can be replaced with that of an ideal, a subset of the elements of a ring that contains all sums of pairs of its elements, and all products of its elements with ring elements.