Let N = {1,2,3,} denote the set of natural numbers. It is a well known result from elementary number theory that
each natural number z (which does not equal 1) is a unique product (up to the order) of prime numbers. A number
p is called a prime number if 1 and p are the only divisors of p within N. This means that p cannot be decomposed
into a (nontrivial) product of natural numbers. We rather call numbers with that property "irreducible" instead of
"prime". Moreover, we call any decomposition of an element z into irreducible elements a factorization of z. The
main theorem of elementary number theory hence states, that every natural number has a unique factorization.
The above definition of "irreducible" makes sense in a much more general context than just for natural numbers:
For all mathematical structures where a "multiplication" is defined, we can speak about divisors of an element and
we hence are able to define irreducible elements and factorizations. In particular, this is possible for a very
important class of mathematical objects, called rings (this are sets where an addition and a multiplication is
defined, obeying similar rules like those in Z = {,-2,-1,0,1,2,}).
It turns out that many interesting rings R (from algebraic number theory, algebraic geometry,) are atomic, i.e. all
elements of R have factorizations into irreducible elements. However, these factorizations are not unique in
general: each element of R can possess many different factorizations.
The long term objective of factorization theory is an utmost precise description of phenomena of non-uniqueness of
factorizations. The selected research methods in this young area of research include commutative algebra,
idealtheory, semigroup theory, combinatorial methods, additive number theory, group theory, algebraic geometry
and even analytical methods.
Before arithmetical investigations of certain classes of rings become possible, extensive algebraic knowledge about
these mathematical objects is indispensable. Hence the study of their algebraic properties takes a large amount of
research work. And the results obtained there are not only useful for factorization theory: they lead to more
comprehensive algebraic insight into the structure of these mathematical objects.