# Semiring of classes of associates of a ring/rng

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:

• $$[a] \cdot [b] = [a \cdot b]$$

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 $$\langle a \rangle + \langle b \rangle$$.

• You are making a conjecture here. Before making a conjecture, it is always a good idea to look at some examples. Have you considered any examples relating to your conjecture? – Rob Arthan Feb 26 at 23:30
• @RobArthan I checked the ring of integers (the example is in the question). Currently checking cyclic rings. – Alex C Feb 26 at 23:33