Why prior in MAP could be ignored?
The maximum a posteriori definition usually goes like this:

$$ \arg\max_{\theta} p(\theta|x) = \arg\max_{\theta} \frac{p(x|\theta)\cdot p(\theta)}{p(x)} $$

and given that $p(x)$ is independent of $\theta$, it's not needed for finding the $\arg\max_{\theta}$, then you have 

$$ \arg\max_{\theta} p(\theta|x) = \arg\max_{\theta} \frac{p(x|\theta)\cdot p(\theta)}{p(x)} =  \arg\max_{\theta} p(x|\theta)\cdot p(\theta)$$