In my question, I linked to a sci.math.research archive that is now defunct, so I am re-posting (most of) that content here.

Phil wrote:

What's the Grothendieck completion of this semiring? That might be something interesting.

Olivier wrote:

$I+J$ is linked with gcd—these beasts do not have a good structure,
but, but, some investigations is surely called for. One usually sees
the ordered lattice with $I \subset J$ and $I+J$ and $I \cap J$. It seems
it behaves nicely with respect to multiplication, so one should be
able to prove something akin to "this lattice is a product lattice
over all prime ideals". This result in itself would not be very
sexy. However, when it *fails* would most probably be more attractive.
The distinction between maximal and prime ideal is of major incidence
in ring theory, and it is clear such a distinction will have impact
on the core "result" I mentioned.

Pete Clark wrote:

By coincidence a student asked me the same question about a month
ago. It seemed promising at first, until we noticed that for all
ideals $I$, $I + I = I$. Therefore $I = 0$ in the ring completion—i.e.,
the completion is the zero ring.

I responded:

That's a good point. I'm not quite ready to give up yet, though. There
does seem to be some interesting structure. For example, in the integers,
multiplication is multiplication, and addition is gcd. I think that the
"ideal semiring" of any Dedekind domain is isomorphic to that of the
integers. In general, though, one gets something else—I'm not sure
what.

Experience with tropical semirings suggests that simply trying to complete
a semiring to a ring isn't always a good idea. I don't really know what
tools are appropriate here though.

Pete Clark pointed out that my statement about Dedekind domains was wrong:

No, that's not right. For instance, if $R$ is the univariate polynomial
ring over the complex numbers, then its ideal semiring is uncountable.

Perhaps what you meant to say is the following? The ideal semiring of
any Dedekind domain $R$ is isomorphic to
the direct sum of copies of the semiring $(\mathbb{N} \cup \{\infty\},\min,+)$, where $\mathbb{N}$ is
the natural numbers (positive integers plus 0),
the addition law is given by the minimum, and the multiplication law
is given by addition. The direct sum extends over
all nonzero prime ideals of $R$. (And yes, this is strongly reminsicent
of tropical geometry….)

Agreed that for a non-Dedekind domain the structure will be much
different. As someone else already noted, it has a lattice-theoretic
feel to it, but I don't know what work has been done in this
direction.

I wrote:

I asked a friend of mine this question and one of his first instincts was
to ask, when is this semiring finitely generated?

The answer might be "not very often" since even the integers don't give an
example, but if there are some nontrivial examples, these might be the most
tractable to analyze.

Dave Cullen wrote (I hope I'm interpreting his $*$ notation correctly; originally he wrote `J*`

and `A*B`

and `A*X`

):

Well, one thing I notice from this conversation is that the set of
ideals of a ring with ideal sum and multiplication does form a
complete (idempotent) dioid; in particular, the operation $*$, given as
$$J^* = \sum_{i\in \mathbb{N}} J^i$$
is well-defined for an ideal $J$. Such structures have a partial
ordering "$\le$" given by
$$J \le K \iff J + K = K.$$
Some preliminary results are that for fixed ideals $A$ and $B$,
the affine equation $X = AX + B$ has a least (with respect to $\le$)
solution $A^*B$, and any solution $X$ satisfies $X = A^*X$.

It seems like these dioids are well studied in other contexts, but I
have never actually seen anything (in the web-literature anyhow) about
dioids of ideals. There is also something called a cost dioid that is
well studied, again for motivating reasons totally unrelated to sets
of ideals, which requires the additional condition of the existence of
roots of elements. Although this condition fails for general ideals,
it may be true for some rings, or for some subsets of the collection
of ideals of a ring. Maybe we can steal some of this dioid theory and
apply it to the ideal context? Comments welcome!

whichproperties can be covered? $\endgroup$