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

Alex C
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  • 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 '20 at 23:30
  • @RobArthan I checked the ring of integers (the example is in the question). Currently checking cyclic rings. – Alex C Feb 26 '20 at 23:33

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