I want to know some examples with the following properies.
Let $R$ be a domain such that every non unit element $x$ is a product of finite irreducible elements, but $R$ is not a UFD, and there is some element $y\in R$ such that $y$ has two distinct factorizations with different lengths.
Textbooks tell me, $\mathbb{Z}[\sqrt{-5}]$ is a non-UFD since $6=2\cdot3=(1+\sqrt{-5})(1-\sqrt{-5})$.
Since $\mathbb{Z}[\sqrt{-5}]$ is Noetherian, then it is easy to show that every non unit element is a product of finite irreducible elements.
But I donot know if every two factorizations of any given element of $\mathbb{Z}[\sqrt{-5}]$ have the same lengths ? That is to ask if this is an example? More, what about general algebraic integer domains ?
What is the famous (easy understood) example that a atomic domain is not a HFD (any two factorizations of any given $x$ have the same length)?
Thanks.
Best Answer
Take $R$ to be the ring of polynomials in which the $x$ term does not appear, i.e. polynomials of the form $$a_0+a_2x^2+...$$ Then by induction on the degree we have that $R$ is an atomic domain, since every element is prime or product of two polynomials with less degree. And if we take the polynomial $$x^6$$ we have that $$x^2\cdot x^2 \cdot x^2$$ and $$x^3\cdot x^3$$ have different length as desired. (One can easily prove that $x^2$ and $x^3$ are prime in $R$.)