Integral domain such that.....

[wildrabbitt]

Hi,


Let R be an integral domain in which every non-zero non-unit element has a factorisation into irreducible elements.

Can someone give me a concrete example of such a Ring?

[Nick]

Hi,
Let R be an integral domain in which every non-zero non-unit element has a factorisation into irreducible elements.

Can someone give me a concrete example of such a Ring?

The set of all integers (under ordinary addition and multiplication) is an integral domain. Its units are 1 and -1, and its irreducible elements are:
2,-2,3,-3,5,-5,7,-7,11,-11,13,-13,17,-17,...
i.e. the prime numbers together with their corresponding negative numbers.
As every integer greater than 1 can be written as a product of prime numbers and every integer less than -1 too when we involve the negative versions, it follows that this is an example.

The set of all real numbers is also an example, but a less interesting one, since every element except 0 is a unit!

[wildrabbitt]

Big thanks.