What does it mean for a factorization to be unique up to unit factors?

Question

"If $R$ is a UFD, then the factorization of any element as a finite product of irreducible factors is unique up to within order and unit factors."

One could argue that this means that if for instance $ab$ and $cd$ are factorizations of a number into irreducible element, then there is a unit $u$ such that $au = c$ and $b$ = $d$. This seems fishy too me.

I think it means that if for instance $ab$ and $cd$ are factorizations of a number into irreducible elements, then there are units $u$ and $v$, with $uv = 1$ such that $au = c$ and $bv = d$. The reason beeing that given a factorization one could multiply by $1 = uv$ and then multiply $u$ with one irreducible element and $v$ with another, thus maybe changing the factors.

What is the correct way to interpret the statement?

Edit: Changed the title

Answer 1

You're exactly right. Let $R$ be a UFD, then you could state the unique factorization condition as follows:

For any $a\in R$, let $[a]$ denote the set $\{au : u\in R^\times\}$.

Then if $a_1a_2\cdots a_n = b_1b_2\cdots b_m$ with $a_i,b_i$ irreducible, then $n = m$, and there is some permutation $\sigma\in S_n$ such that $[a_i] = [b_{\sigma(i)}]$ for every $i$.