Let's call elements that generate the same principal ideal of a ring/rng associates.
$[a]$ is the equivalence class of associates of an element $a$ of a ring/rng.
Let $A = \{[a],[b], ...\}$ be the set of all classes of associates of a ring/rng.
We can define the multiplication on $A$ in a commutative ring in the following way:
- $[a] \cdot [b] = [a \cdot b]$
Product of principal ideals: $(a)\cdot (b) = (a b)$.
It looks like if we define the addition for the ring of integers as
- $[a] + [b] = [|a| + |b|]$,
the structure will be a semiring
(since the operations on the classes are equivalent to operations on absolute values in $\mathbb Z$).
Is it possible to define the addition
- $[a] + [b] = [c]$
in a way that the structure will be a semiring for an arbitrary commutative ring/rng?
Is it possible for a certain class of rings (e.g. principal ideal rings/rngs)?
Are there any interesting properties of such a structure?
Update
It looks like the set of classes of associates is a semiring in a principal ideal ring/rng
since ideals of a ring form a semiring: https://mathoverflow.net/q/26607/148743
Then, $[a] + [b] = $ the class of generators of $\langle a \rangle + \langle b \rangle$.