How to multiply and reduce ideals in quadratic number ring. I am studying quadratic number rings and I have a problem with multiplying and reducing ideals, for example:

Let $w=\sqrt{-14}$. Let $a=(5+w,2+w)$, $b=(4+w,2-w)$ be ideals in $\mathbb Z[w]$. Now, allegedly, the product of ideals $a$ and $b$ in $\mathbb Z[w]$ is $(6,3w)$. Please explain clearly, how to get to $(6,3w)$.


You reduce an ideal by adding a multiple of one generator to another with the goal of simplifying things (making the numbers smaller and removing appearances of $w$)
First, reduce $a$ and $b$ :
$a = (5+w,2+w) = (3,2+w) \\ b = (4+w,2-w) = (6,2-w)$
Then, multiply every generator of $a$ with veery generator of $b$, and remove the $w^2$ terms :
$ab = (18, 6-3w,12+6w,4-w^2) = (18,6-3w,12+6w,18)$
Then reduce :
$ab = (18,6-3w,12+6w) = (18,6-3w,24) = (18,6-3w,6) = (6,3w)$
