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 $R$ having a divisor theory in the following sense: there exists a monoid D $D$ (i.e., a commutative semigroup with cancellation and a unit element) with the GCD-property for the associated group G $G$ of quotients, and a homomorphism div$\mathrm{div}$ of the multiplicative group K^* $K^*$ of the quotient field of R $R$ into G $G$ with the following two properties:
(i) If a,b in K* $a,b \in K^*$ and b/a in R $b/a \in R$, then div(b)/div(a) in D $\mathrm{div}(b)/\mathrm{div}(a) \in D$, and
(ii) for every element d in D $d \in D$ there exists a set A < K* $A \subseteq K^*$
such that d $d$ is the gcd of {div(a) : a in A} $\{\mathrm{div}(a) : a \in A\}$.
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
