The following wonderful 54 page survey by O. Neumann on Kronecker's divisor theory could easily be turned into a book and would fill a very large gap in the English literature on such. I'm interested in helping if anyone is game for such (but, alas, my German is weak).

Neumann, O.(D-FSU-MI) 2003k:13021 13F05 (01A55 13G05 20M14)

Was sollen und was sind Divisoren? (German. German summary)

[What are divisors and what can we do with them?]

Math. Semesterber. * 48 (2002), no. 2, 139--192.

In the first part of this paper a survey is given of the development of
Kronecker's theory of divisors. In the second part the author develops a
theory of integral domains $R$ having a divisor theory in the following sense:
there exists a monoid $D$ (i.e., a commutative semigroup with cancellation and
a unit element) with the GCD-property for the associated group $G$ of
quotients, and a homomorphism $\mathrm{div}$ of the multiplicative group $K^*$ of the
quotient field of $R$ into $G$ with the following two properties:

(i) If $a,b \in K^*$ and $b/a \in R$, then $\mathrm{div}(b)/\mathrm{div}(a) \in D$, and

(ii) for every element $d \in D$ there exists a set $A \subseteq K^*$

such that $d$ is the gcd of $\{\mathrm{div}(a) : a \in A\}$.

The author states that a similar theory was presented in the thesis of
F. Lucius ["Ringe mit einer Theorie des groessten gemeinsamen Teilers", Ph.D.
thesis, Univ. Gottingen, Gottingen, 1996; Zbl 0901.13002]. After developing
the fundamental properties of such divisor theory, relations to the approaches
of Kronecker, Zolotarev and Dedekind are established.

--Reviewed by W. Narkiewicz