Remark

There also exist arithmetic systems in which uniqueness of factorizations is not guaranteed.

For example, in the system Z[(-5)] of numbers of the form a+b(-5) with a, b Z we can express 6 in two essentially different ways:

6 = 3 · 2 = (1+(-5)) · (1-(-5)).

The system Z[(-5)] is an example of a ring, a structure we will encounter in Chapter 7.