How to compute the product of ideals Let $R=\mathbb{Z}[\sqrt{-105}]$. Consider the following ideals in $R$:


*

*$\mathfrak{p}_2=(2,1+\sqrt{-105})$

*$\mathfrak{p}_5=(5,\sqrt{-105})$

*$\mathfrak{p}_{13}=(13,5+\sqrt{-105})$
I know the definition of product of ideals, but it seems not so useful for concrete computations. How can I find, for example, the product $2\mathfrak{p}_2\mathfrak{p}_5\mathfrak{p}_{13}$?

Let $I=(x_1, x_2)$, and $J=(y_1, y_2)$. Take the definition, for a commutative ring, that $I=\{r_1x_1+r_2x_2|r_1, r_2 \in R\}, J=\{s_1y_1+s_2y_2|s_1, s_2 \in R\}$. Then take the product of these ideals, producing a description of it as a quantified set, and throw out terms from $R$ that don't add any extra elements. You will get a more feasible description of $IJ$. You can generalize this to ideals with more generators.
Beyond this, simplifying the description is a matter of finding a basis, a generating set of minimum size. For this, you can use Groebner bases, Wu's method of characteristic set, or factorizing by inspection say with Horner's method, or other techniques, although these are more for polynomial factorization, and I haven't tried them on other commutative rings.
