Return to Question

In [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ascending chain condition on principal ideals (ACCP). Some years later, A. Grams [Math. Proc. Cambridge Phil. Soc. 75 (1974), No. 3, 321-329] showed (by way of a counterexample) that Cohn's statement is wrong: Every commutative domain satisfying the ACCP is atomic, but not the other way around. It is therefore natural to wonder if Cohn's claim can be, in a sense, fixed by providing a sensible characterization (say, of a somewhat ideal-theoretic flavour) of when a commutative domain (or, more generally, a cancellative commutative monoid) is atomic.

Edit 2 (addressing a comment by მამუკა ჯიბლაძე). Apart from Grams' counterexample to Cohn's statement, further examples of atomic commutative domains that do not satisfy the ACCP were given by A. Zaks [J. Algebra 80 (1982), 223-231] (where the author considers certain quotients of a polynomial ring in infinitely many variables and proves that they are atomic without the ACCP) and M. Roitman [J. Pure Appl. Algebra 87 (1993), 187-199] (where, in Example 5.1, the author famously shows the existence of an atomic commutative domain $R$ such that the univariate polynomial ring $R[X]$ is not atomic, incidentally producing an atomic commutative domain without the ACCP). More recently, further examples were provided by J. G. Boynton and J. Coykendall [J. Pure Appl. Algebra 223 (2019), 619-625] (where the authors use pullbacks of commutative rings to construct large families of atomic commutative domains without the ACCP) and F. Gotti and B. Li [https://arxiv.org/abs/2111.00170] (where, among other things, the authors construct what appears to be the first-ever example of an atomic commutative monoid ring that does not satisfy the ACCP).

It is definitely much easier to construct cancellative commutative monoids without the ACCP. For instance, S.T. Chapman et al. prove in Corollary 4.4 of [Amer. Math. Monthly 128 (2021), No. 4, 302-321] that, if $r$ is a non-zero rational number smaller than $1$ whose numerator is not $1$, then the cyclic Puiseux monoid generated by $r$ (i.e., the submonoid of the additive group of the rational field generated by $1, r, r^2, \ldots$) is atomic but does not satisfy the ACCP.

Let $R$ be a commutative domain. As remarked by Cohn himself, all the notions above are described by the partially ordered set $L_p(R)$ of all principal ideals of $R$. For instance, $R$ satisfies the ACCP if and only if $L_p(R)$ is a noetherian poset, and $R$ satisfies the DCCP if and only if $L_p(R)$ is artinian. It is easily seen that $R$ is atomic if and only if, for every $I \in L_p(R)$, the interval $$ [I,R] :=\{\, J\in L_p(R) \colon I \subseteq J\subseteq R\,\} $$ has a maximal finite chain. Here, a maximal finite chain of the interval $[I,R]$ is a finite chain $I = I_0 \subsetneq I_1 \subsetneq \cdots \subsetneq I_n = R$ of $[I,R]$ that cannot be properly refined into any other finite chain of $[I,R]$. See [A. Facchini and M. Fassina, Factorization of elements in noncommutative rings, II, Comm. Algebra 46 (2018), No. 7, 2928-2946]." Alberto adds, "I don't know if that counts as a satisfactory ideal-theoretic characterization."

In Proposition 1.1 of [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ascending chain condition on principal ideals (ACCP). Some years later, A. Grams [Math. Proc. Cambridge Phil. Soc. 75 (1974), No. 3, 321-329] showed (by way of a counterexample) that Cohn's statement is wrong: Every commutative domain satisfying the ACCP is atomic, but not the other way around.

Grams' construction is commonly acknowledged as the first counterexample to Cohn's claim; and until now (2022-08-14, 13:40 CET), I too have believed that this was indeed the case. However, it appears that Cohn himself had already realized his mistake and sketched a (somewhat easier) counterexample on p. 4, ll. 14-18 of [Amer. Math. Monthly 80 (Jan., 1973), No. 1, 1-18].

In any case, it is natural to wonder if Cohn's claim can be, in a way, fixed by providing a sensible characterization (say, of a somewhat ideal-theoretic flavour) of when a commutative domain (or, more generally, a cancellative commutative monoid) is atomic.

Edit 2 (addressing a comment by მამუკა ჯიბლაძე). Apart from Grams' and Cohn's counterexamples (to Cohn's statement), atomic commutative domains that do not satisfy the ACCP are constructed by A. Zaks in [J. Algebra 80 (1982), 223-231] (where the author considers certain quotients of a polynomial ring in infinitely many variables and proves that they are atomic without the ACCP) and by M. Roitman in [J. Pure Appl. Algebra 87 (1993), 187-199] (where, in Example 5.1, the author famously shows the existence of an atomic commutative domain $R$ such that the univariate polynomial ring $R[X]$ is not atomic, incidentally producing an atomic commutative domain without the ACCP). More recently, further examples were provided by J. G. Boynton and J. Coykendall [J. Pure Appl. Algebra 223 (2019), 619-625] (where the authors use pullbacks of commutative rings to construct large families of atomic commutative domains without the ACCP) and F. Gotti and B. Li [https://arxiv.org/abs/2111.00170] (where, among other things, the authors construct what appears to be the first-ever example of an atomic commutative monoid ring that does not satisfy the ACCP).

It is definitely much easier to construct cancellative commutative monoids without the ACCP. For instance, S.T. Chapman et al. prove in Corollary 4.4 of [Amer. Math. Monthly 128 (2021), No. 4, 302-321] that, if $r$ is a non-zero rational number smaller than $1$ whose numerator is not $1$, then the additive monoid of the cyclic Puiseux semiring generated by $r$ (i.e., the submonoid of the additive group of the rational field generated by $1, r, r^2, \ldots$) is atomic but does not satisfy the ACCP.

Let $R$ be a commutative domain. As remarked by Cohn himself, all the notions above are described by the partially ordered set $L_p(R)$ of all principal ideals of $R$. For instance, $R$ satisfies the ACCP if and only if $L_p(R)$ is a noetherian poset, and $R$ satisfies the DCCP if and only if $L_p(R)$ is artinian. It is easily seen that $R$ is atomic if and only if, for every $I \in L_p(R)$, the interval $$ [I,R] :=\{\, J\in L_p(R) \colon I \subseteq J\subseteq R\,\} $$ has a maximal finite chain. Here, a maximal finite chain of the interval $[I,R]$ is a finite chain $I = I_0 \subsetneq I_1 \subsetneq \cdots \subsetneq I_n = R$ of $[I,R]$ that cannot be properly refined into any other finite chain of $[I,R]$. See [A. Facchini and M. Fassina, Factorization of elements in noncommutative rings, II, Comm. Algebra 46 (2018), No. 7, 2928-2946]." I don't know though if that counts as a satisfactory ideal-theoretic characterization."

Edit 3. Let me share the following excerpt from an exchange with Alberto Facchini (I'm doing so with his permission): It doesn't really answer my question, but seems worth mentioning.

Let $R$ be a commutative domain. As remarked by Cohn himself, all the notions above are described by the partially ordered set $L_p(R)$ of all principal ideals of $R$. For instance, $R$ satisfies the ACCP if and only if $L_p(R)$ is a noetherian poset, and $R$ satisfies the DCCP if and only if $L_p(R)$ is artinian. It is easily seen that $R$ is atomic if and only if, for every $I \in L_p(R)$, the interval $$ [I,R] :=\{\, J\in L_p(R) \colon I \subseteq J\subseteq R\,\} $$ has a maximal finite chain. Here, a maximal finite chain of the interval $[I,R]$ is a finite chain $I = I_0 \subsetneq I_1 \subsetneq \cdots \subsetneq I_n = R$ of $[I,R]$ that cannot be properly refined into any other finite chain of $[I,R]$. See [A. Facchini and M. Fassina, Factorization of elements in noncommutative rings, II, Comm. Algebra 46 (2018), No. 7, 2928-2946]." Alberto adds, "I don't know if that counts as a satisfactory ideal-theoretic characterization."

Let me recall that a (multiplicatively written) monoid $H$ is atomic if every non-unit factors as a product of atoms (i.e., elements that cannot be written as a product of two non-units); and it satisfies the ACCP if there does not exist any (infinite) sequence $x_1, x_2, \ldots$ in $H$ such that $Hx_i H \subsetneq Hx_{i+1} H$ for all $i \in \mathbb N^+$. Accordingly, a domain is atomic if so is the multiplicative monoid $R^\bullet$ of its non-zero elements; and satisfies the ACCP if so does $R^\bullet$.

Edit 2 (addressing a comment by მამუკა ჯიბლაძე). Apart from Grams' counterexample to Cohn's statement, further examples of atomic commutative domains that do not satisfy the ACCP were given by A. Zaks [J. Algebra 80 (1982), 223-231] (where the author considers certain quotients of a polynomial ring in infinitely many variables and proves that they are atomic without the ACCP) and M. Roitman [J. Pure Appl. Algebra 87 (1993), 187-199] (where the author famously shows the existence of an atomic commutative domain $R$ such that the polynomial ring $R[X]$ is not atomic, incidentally producing an atomic commutative domain without the ACCP). More recently, further examples were provided by J. G. Boynton and J. Coykendall [J. Pure Appl. Algebra 223 (2019), 619-625] (where the authors use pullbacks of commutative rings to construct large families of atomic commutative domains without the ACCP) and F. Gotti and B. Li [https://arxiv.org/abs/2111.00170] (where, among other things, the authors construct what is probably the first-ever example of an atomic commutative monoid ring that does not satisfy the ACCP).

It is definitely much easier to construct cancellative commutative monoids without the ACCP. For instance, S.T. Chapman et al. prove in Corollary 4.4 of [Amer. Math. Monthly 128 (2021), No. 4, 302-321] that, if $r$ is a non-zero rational number smaller than $1$ whose numerator is not $1$, then the cyclic Puiseux monoid generated by $r$ (i.e., the submonoid of the additive group of the rational field generated by $1, r, r^2, \ldots$) is atomic but does not satisfy the ACCP.

Let me recall that a (multiplicatively written) monoid $H$ is atomic if every non-unit factors as a product of atoms (i.e., non-units that cannot be written as a product of two non-units); and it satisfies the ACCP if there does not exist any (infinite) sequence $x_1, x_2, \ldots$ in $H$ such that $Hx_i H \subsetneq Hx_{i+1} H$ for all $i \in \mathbb N^+$. Accordingly, a domain is atomic if so is the multiplicative monoid $R^\bullet$ of its non-zero elements; and satisfies the ACCP if so does $R^\bullet$.

Edit 2 (addressing a comment by მამუკა ჯიბლაძე). Apart from Grams' counterexample to Cohn's statement, further examples of atomic commutative domains that do not satisfy the ACCP were given by A. Zaks [J. Algebra 80 (1982), 223-231] (where the author considers certain quotients of a polynomial ring in infinitely many variables and proves that they are atomic without the ACCP) and M. Roitman [J. Pure Appl. Algebra 87 (1993), 187-199] (where, in Example 5.1, the author famously shows the existence of an atomic commutative domain $R$ such that the univariate polynomial ring $R[X]$ is not atomic, incidentally producing an atomic commutative domain without the ACCP). More recently, further examples were provided by J. G. Boynton and J. Coykendall [J. Pure Appl. Algebra 223 (2019), 619-625] (where the authors use pullbacks of commutative rings to construct large families of atomic commutative domains without the ACCP) and F. Gotti and B. Li [https://arxiv.org/abs/2111.00170] (where, among other things, the authors construct what appears to be the first-ever example of an atomic commutative monoid ring that does not satisfy the ACCP).

It is definitely much easier to construct cancellative commutative monoids without the ACCP. For instance, S.T. Chapman et al. prove in Corollary 4.4 of [Amer. Math. Monthly 128 (2021), No. 4, 302-321] that, if $r$ is a non-zero rational number smaller than $1$ whose numerator is not $1$, then the cyclic Puiseux monoid generated by $r$ (i.e., the submonoid of the additive group of the rational field generated by $1, r, r^2, \ldots$) is atomic but does not satisfy the ACCP.