Principal ideal domains are thus mathematical objects which
behave somewhat like the integers, with respect to divisibility:
any element of a PID has a unique decomposition into prime
elements (so an analogue of the fundamental theorem of
arithmetic holds); any two elements of a PID have a greatest
common divisor. A principal ideal domain is a specific type of
integral domain, and can be characterized by the following (not
necessarily exhaustive) chain of class inclusions: Examples of
integral domains that are not PIDs: It is not principal because
the ideal generated by 2 and X is an example of an ideal that
cannot be generated by a single polynomial. If M is a free
module over a principal ideal domain R, then every submodule of
M is again free. This does not hold for modules over arbitrary
rings, as the example of modules over shows. All Euclidean
domains are principal ideal domains, but the converse is not
true. An example of a principal ideal domain that is not a
Euclidean domain is the ring . Every princi