***Note:** Please let me know if this question is too basic for MathOverflow. It is about a subject commonly taught in graduate school (commutative algebra), and is based in large part on a [(very elementary) research paper from 2018](http://www.gioenia.unict.it/bollettino/bollettino2018/Full_Papers/Arithmetic%20Rings%20and%20their%20generalizations.pdf), and seems to be about a concept ("super-arithmetic" rings) which may not have been studied in the literature, so it seemed like it might fit, but I wasn't sure. If the question is too basic, please let me know so I can close and delete it and re-ask it on Math.SE.*


Let $\mathfrak{a, b, a\_i, b\_i}$ denote ideals of an arithmetic ring $R$.



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> **Question:** Is it *always* true that $\sum\_i\mathfrak{a}\cap \mathfrak{b\_i=a\cap(\sum\_ib\_i)}$ and $\mathfrak{a(\bigcap\_i b\_i)=\bigcap\_iab\_i}$ in arithmetic rings, even for infinite sums or infinite intersections? If not, what are some counterexamples?
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Since one has that $\mathfrak{a \cap b\_1 + a\cap b\_2 = a \cap (b\_1 + b\_2)}$ and $\mathfrak{a(b\_1 \cap b\_2) = ab\_1 \cap a b\_2}$ for arithmetic rings, it follows by associativity of addition of ideals and of intersection, plus induction, that the above equalities *are* always true for *finite* intersections and *finite* sums. But of course such a proof technique doesn't extend to the infinite case. I suspect the result is false in the infinite case.



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> **Bonus Question 1:** If the above is *not* true in arithmetic rings for *infinite* sums and intersections, then what is the term for rings for which it does hold for index sets of arbitrary cardinality? "Super-arithmetic" rings? Can anyone point to a reference?
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> **Bonus Question 2:** If "super-arithmetic" rings are a strict subset of arithmetic rings, is it true that for "super-arithmetic" rings $\mathfrak{\sum\_i (a\_i : b) = (\sum\_i a\_i : b)}$ and $\mathfrak{\sum\_i (a : b\_i) = (a: \bigcap\_i b\_i)}$ for ideals $\mathfrak{b, b\_i}$ which are infinitely generated and/or for infinite sums?
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How do the answers to these questions relate to localization and/or Noetherian conditions? For example, is it the case that any Noetherian arithmetic ring is also "super-arithmetic", since for any localization $R\_{\mathfrak{m}}$ of $R$ by a maximal ideal $\mathfrak{m}$, all ideals are contained in an ascending chain ($A\_{\mathfrak{m}}$ being [uniserial](https://en.wikipedia.org/wiki/Serial_module) for any maximal $\mathfrak{m}$ is another equivalent characterization of arithmetic rings, see again [here](http://www.gioenia.unict.it/bollettino/bollettino2018/Full_Papers/Arithmetic%20Rings%20and%20their%20generalizations.pdf)), and because $R$ is Noetherian that chain is finite, so all infinite sums and intersections are actually equal to finite sums and intersections (I think? I'm being sloppy obviously).



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> **Bonus Question 3:** Is there any example of a "super-arithmetic" ring which is *not* Noetherian?
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Also I guess the ideal quotient properties aren't likely to apply even in "super-arithmetic" rings when the $\mathfrak{b, b\_i}$ aren't finitely generated since their being finitely generated is the most common sufficient condition for their ideal quotients localizing of which I'm aware.


**Definitions:** An [arithmetic ring](https://en.wikipedia.org/wiki/Arithmetical_ring) is a "Prüfer domain" that isn't necessarily an integral domain. A [Prüfer domain](https://en.wikipedia.org/wiki/Pr%C3%BCfer_domain) is an arithmetic ring which is also an integral domain.


Arithmetic rings are characterized by the property that:


For all ideals $\mathfrak{a, b\_1, b\_2}$,
$$\mathfrak{a \cap (b\_1 + b\_2) = a \cap b\_1 + a \cap b\_2} \,.$$
(I.e. the modular law holds with equality, i.e. $\mathfrak{a \cap b\_1 + a \cap b\_2 \supseteq a \cap (b\_1 + b\_2)}$ since the other direction is always true for arbitrary rings.)


In Prüfer domains (i.e. when the ring can be assumed to be an integral domain) the above characterizing property is equivalent to the following:


For all ideals $\mathfrak{a, b\_1, b\_2}$,
$$ \mathfrak{a (b\_1 \cap b\_2) = ab\_1 \cap ab\_2} \,.$$
(I.e. $\mathfrak{ab\_1 \cap ab\_2 \subseteq a(b\_1 \cap b\_2)}$, since the other direction is always true for arbitrary rings.) However in the full generality of arithmetic rings (i.e. not necessarily integral domains) this property is strictly weaker/more general than the above property ([see this paper for details](http://www.gioenia.unict.it/bollettino/bollettino2018/Full_Papers/Arithmetic%20Rings%20and%20their%20generalizations.pdf)).



Does $\sum_ia\cap b_i=a\cap(\sum_ib_i)$ and $a(\bigcap_i b_i)=\bigcap_iab_i$ for infinite sums and intersections in arithmetic rings (Prufer domains)?
The first identity holds in an arithmetical ring since the identity holds iff it holds locally and an arithmetical ring is locally a chained ring and its easy to see that the identity holds in a chained ring.


