Ideals over a ring : unique factorization property I read a report about rings of quadratic integers $R$. This report explains that the sets $\mathcal I_R$ of ideals of $R$, (with $\times$ defined by $a\times b$ as the ideal generated by $a_i\cdot b_j$ where $a_i$ are the generators of $a$, $b_j$ the generators of $b$ and $\cdot$ the multiplication of $R$) has the property of unique factorization, even if $R$ has not.
Does $\mathcal I_R$ have always the unique factorization property ? What properties of $R$ are needed for that ?
Is it possible to extend $I_R$ with an operation $+$ to obtain a ring ? a principal ideal domain ?
Any reference about that subject is welcome.
 A: 
What properties of $R$ are needed for that?

This is a key feature of Dedekind domains, of which quadratic integers are an example. 

Does $\mathcal I_R$ have always the unique factorization property?

Certainly not. The property is very special. Even the ideals of $\mathbb Q\times \mathbb Q$ do not satisfy unique factorization. Among domains I think unique factorization of ideals characterizes Dedekind domains.

Is it possible to extend $\mathcal I_R$ with an operation $+$ to obtain a ring?

It depends on the monoid structure of $\mathcal I_R$ whether or not there is some random addition that makes it into a ring. 
"Some random addition" isn't very interesting through since it's probably not related to $R$ at all. Now, there is already a natural binary operation to combine ideals with addition, so it is natural to ask how well that does. While ideal addition is commutative and associative and has an identity, it fails to have inverses because $I+I=I$ for every ideal. But this still gives $\mathcal I_R$ the interesting structure of a semiring. Every ring has an associated semiring of ideals.
To make the semiring "domain-like" you would have to ask under what conditions $AB=0$ implies $A=0$ or $B=0$ for a pair of ideals $A,B$. This is equivalent to $R$ being an integral domain, so it is true for the ideal semiring of any integral domain.
I can't speak to when the semiring's ideals are principal. I am less familiar with that particular aspect of their structure.
