When does the distributive law apply to ideals in a commutative ring?

Question

Let $R$ be a commutative ring with identity and $I,J,K$ be ideals of $R$. If $I\supseteq J$ or $I\supseteq K$, we have the following modular law $$ I\cap (J+K)=I\cap J + I\cap K$$

I was wondering if there are situations in which the modular law holds in which the hypothesis that $I$ contains at least one of $J,K$ is relaxed. (This amounts to the lattice of ideals being distributive.)

One example is when $R$ is a polynomial ring or power series ring and $I,J,K$ are monomial ideals.

Of course one containment always holds $I\cap (J+K)\supseteq I\cap J +I\cap K$. In what other situations does the other containment hold?

Answer 1

Such domains are known as Prüfer domains. They are non-Noetherian generalizations of Dedekind domains. Their ubiquity stems from a remarkable confluence of interesting characterizations. For example, they are those domains satisfy either the Chinese Remainder Theorem for ideals, or Gauss's Lemma for polynomial content ideals, or for ideals: $\rm\ A\cap (B + C) = A\cap B + A\cap C\:,\ $ or $\rm\ (A + B)\ (A \cap B) = A\ B\:,\ $ or $\rm\ A\supset B\ \Rightarrow\ A\:|\:B\ $ for fin. gen. $\rm\:A\:$ etc. It's been estimated that there are close to 100 such characterizations known, e.g. see my sci.math post for 30 odd characterizations. Below is an excerpt:

THEOREM $\ \ $ Let $\rm\:D\:$ be a domain. The following are equivalent:

(1) $\rm\:D\:$ is a Prüfer domain, i.e. every nonzero f.g. (finitely generated) ideal is invertible.
(2) Every nonzero two-generated ideal of $\rm\:D\:$ is invertible.
(3) $\rm\:D_P\:$ is a Prufer domain for every prime ideal $\rm\:P\:$ of $\rm\:D.\:$
(4) $\rm\:D_P\:$ is a valuation domain for every prime ideal $\rm\:P\:$ of $\rm\:D.\:$
(5) $\rm\:D_P\:$ is a valuation domain for every maximal ideal $\rm\:P\:$ of $\rm\:D.\:$
(6) Every nonzero f.g. ideal $\rm\:I\:$ of $\rm\:D\:$ is cancellable, i.e. $\rm\:I\:J = I\:K\ \Rightarrow\ J = K\:$
(7) $ $ Restriction of (6) to f.g. $\,\rm J,K.$
(8) $\rm\:D\:$ is integrally closed and there is an $\rm\:n > 1\:$ such that for all $\rm\: a,b \in D,\ (a,b)^n = (a^n,b^n).$
(9) $\rm\:D\:$ is integrally closed and there is an $\rm\: n > 1\:$ such that for all $\rm\:a,b \in D,\ a^{n-1} b \ \in\ (a^n, b^n).$
(10) Each ideal $\rm\:I\:$ of $\rm\:D\:$ is complete, i.e. $\rm\:I = \cap\ I\: V_j\:$ as $\rm\:V_j\:$ run over all the valuation overrings of $\rm\:D.\:$
(11) Each f.g. ideal of $\rm\:D\:$ is an intersection of valuation ideals.
(12) If $\rm\:I,J,K\:$ are nonzero ideals of $\rm\:D,\:$ then $\rm\:I \cap (J + K) = I\cap J + I\cap K.$
(13) If $\rm\:I,J,K\:$ are nonzero ideals of $\rm\:D,\:$ then $\rm\:I\ (J \cap K) = I\:J\cap I\:K.$
(14) If $\rm\:I,J\:$ are nonzero ideals of $\rm\:D,\:$ then $\rm\:(I + J)\ (I \cap J) = I\:J.\ $ ($\rm LCM\times GCD$ law)
(15) If $\rm\:I,J,K\:$ are nonzero ideals of $\rm\:D,\:$ with $\rm\:K\:$ f.g. then $\rm\:(I + J):K = I:K + J:K.$
(16) For any two elements $\rm\:a,b \in D,\ (a:b) + (b:a) = D.$
(17) If $\rm\:I,J,K\:$ are nonzero ideals of $\rm\:D\:$ with $\rm\:I,J\:$ f.g. then $\rm\:K:(I \cap J) = K:I + K:J.$
(18) $\rm\:D\:$ is integrally closed and each overring of $\rm\:D\:$ is the intersection of localizations of $\rm\:D.\:$
(19) $\rm\:D\:$ is integrally closed and each overring of $\rm\:D\:$ is the intersection of quotient rings of $\rm\:D.\:$
(20) Each overring of $\rm\:D\:$ is integrally closed.
(21) Each overring of $\rm\:D\:$ is flat over $\rm\:D.\:$
(22) $\rm\:D\:$ is integrally closed and prime ideals of overrings of are extensions of prime ideals of $\rm\:D.$
(23) $\rm\:D\:$ is integrally closed and for each prime ideal $\rm\:P\:$ of $\rm\:D,\:$ and each overring $\rm\:S\:$ of $\rm\:D,\:$ there is at most one prime ideal of $\rm\:S\:$ lying over $\rm\:P.\:$
(24) For polynomials $\rm\:f,g \in D[x],\ c(fg) = c(f)\: c(g)\:$ where for a polynomial $\rm\:h \in D[x],\ c(h)\:$ denotes the "content" ideal of $\rm\:D\:$ generated by the coefficients of $\rm\:h.\:$ (Gauss' Lemma)
(25) Ideals in $\rm\:D\:$ are integrally closed.
(26) If $\rm\:I,J\:$ are ideals with $\rm\:I\:$ f.g. then $\rm\: I\supset J\ \Rightarrow\ I|J.$ (contains $\:\Rightarrow\:$ divides)
(27) the Chinese Remainder Theorem $\rm(CRT)$ holds true in $\rm\:D\:,\:$ i.e. a system of congruences $\rm\:x\equiv x_j\ (mod\ I_j)\:$ is solvable iff $\rm\:x_j\equiv x_k\ (mod\ I_j + I_k).$
(28) Each finitely generated torsion-free $\rm\,D$-module is projective.

Answer 2

To provide an answer that's closer to the original question and less specialized than the existing answer, I'd like to expand on the comment Matemáticos Chibchas made that these rings are called arithmetical in the literature.

A commutative ring (with identity of course) is called arithmetical if its lattice of ideals is distributive (which is the standard name for the condition you are asking about.)

As it turns out, this is equivalent to every localization $R_M$ of $R$ at a maximal ideal $M$ has linearly ordered ideals, which is the analogue of 5) on the list already given.

And yes, this condition is well-studied for domains, where it defines the class of Prüfer domains. I might add another characterization not exactly on the list given already:

28') a domain whose f.g. ideals are all projective

A commutative ring is called Bézout if every finitely generated ideal is a principal ideal. It is not hard to see that Bézout rings are arithmetical, and therefore so are principal ideal rings. Of course, Bézout domains and PIDs are also very well-studied arithmetical/Prüfer domains.

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I believe arithmetical rings got their start in

Fuchs, Ladislaus. "Über die Ideale arithmetischer ringe." Commentarii Mathematici Helvetici 23.1 (1949): 334-341.

and many things are known about them.

I am not an expert in that literature, but lately I have encountered them while reading about a subclass of so-called "canonical form (CF) rings" in which direct sums of cyclic ideals admit a canonical form à la the canonical form of f.g. modules over PIDs, a natural progression of that theory. Specifically, this is in

Shores, Thomas S., and Roger Wiegand. "Rings whose finitely generated modules are direct sums of cyclics." Journal of Algebra 32.1 (1974): 152-172.

And this in turn is an outgrowth of Kaplansky's classical problem asking "for which rings are all f.g. modules direct sums of cyclic modules?" as in

Kaplansky, Irving. "Elementary divisors and modules." Transactions of the American Mathematical Society 66.2 (1949): 464-491.

Kaplansky, Irving. "Modules over Dedekind rings and valuation rings." Transactions of the American Mathematical Society 72.2 (1952): 327-340.