Suppose $ a, b, c, d $ are nonzero elements in UFD and we can deduce $ a|c $ from $ ab|cd $, then $ d|b $

Question

I came up with this question when I was trying to solve another problem about modules.

Suppose $ a, b, c, d $ are nonzero elements in UFD and we can deduce $ a|c $ from $ ab|cd $, then $ d|b $.

I have posted my solution below.

Answer 1

Write $ a=\prod_{i=1}^{n}p_i^{a_i}, b=\prod_{i=1}^{n}p_i^{b_i}, c=\prod_{i=1}^{n}p_i^{c_i}, d=\prod_{i=1}^{n}p_i^{d_i} $(omit the units), where $ p_i $ are distinct primes. Then $ ab=\prod_{i=1}^{n}p_i^{a_i+b_i} $ and $ cd=\prod_{i=1}^np_i^{c_i+d_i} $ with $ a_i+b_i\leq c_i+d_i $. Since we can always deduce $ a_i\leq c_i $ from above, then $ b_i \geq d_i $, i.e. $ d|b $.