Unique factorization in fields Suppose $A$ is a commutative $R$-algebra and that is also a field. Define:


*

*For $x,y \in A$, say that $x$ divides $y$ iff $xr = y$ for some $r \in R$.

*Call $x,y \in A$ associates iff each divides the other.

*Say that $I \subseteq A$ is an ideal iff $A$ is an additive subgroup of $A$ and $RI \subseteq I$

*etc.


I think this is often the right context in which to view factorization problems in fields, and possibly in other situations where there are "too many units." For example, when $\mathbb{Q}$ is viewed as a $\mathbb{Z}$-algebra, it has a "unique factorization" property whereby every non-zero $q \in \mathbb{Q}$ can be uniquely expressed, up to associates, as product of prime elements of $\mathbb{Z}$ and reciprocals of primes elements of $\mathbb{Z}.$ Essentially the same idea should work if we're thinking about factorization in abelian groups; given a distinguished submonoid, we get a corresponding divisibility relation, notion of associates, etc. and we can try to prove factorization results in this context.
Where can I learn about this kind of thing?

There is a very nice book by Alfred Geroldinger and Franz Halter-Koch which studies factorization from the monoid standpoint.  There is a survey article here: http://www.uni-graz.at/~geroldin/54-non-unique-fact-survey.pdf  By these two, which I assume turned into the book (which is a bit pricey, but your library might have it/ be able to get it for you).  The book is here: https://www.crcpress.com/product/isbn/9781584885764
You might look at the survey article to see if this is something you are interested in.  They also have an extremely extensive bibliography which could help you out!  Hope that helps!
