Does anyone has a simple proof of the following fact:
An integral domain whose every prime ideal is principal is a principal ideal domain (PID).
Here is a proof followed by conceptual elaboration, from my 2008/11/9 Ask an Algebraist post.
Let R be an integral domain. Let every prime ideal in R be principal.
Prove that R is a principal ideal domain (PID)
Below I present a simpler way to view the proof, and some references.
First let’s recall one well-known proof, as presented by P.L. Clark (edited):
Proof$_{\,1}$ $\ $ Suppose not. Then the set of all nonprincipal ideals is nonempty.
Let $\{I_i\}$ be a chain of nonprincipal ideals and put $\,I = \cup_i I_i.\,$ If $I = (x)$ then $x \in I_i$ for some $i,$ so $I = (x) \subset I_i$ implies $I = I_i$ is principal, contradiction.
Thus by Zorn’s Lemma there is an ideal $I$ which is maximal with respect to the
property of not being principal. As is so often the case for ideals maximal
with respect to some property or other, we can show that I must be prime.
Indeed, suppose that $ab \in I$ but neither $a$ nor $b$ lies in I. Then the ideal
$J = (I,a)$ is strictly larger than $I,$ so principal: say $J = (c).$
$I:a := \{r \in R\ :\ ra \in I\}$ is an ideal containing $I$ and $b,$ so strictly larger
than $I$ and thus principal: say $I:a = (d).$ Let $i \in I,$ so $i = uc.$ Now $u(c) \subset I$ so $ua \in I$ so $u \in I:a.$ Thus we may write $u = vd$ and $i = vcd.$ This shows $I \subset (cd).\ $ Conversely, $d \in I:a$ implies $da \in I$ so $d(I,a) = dJ \subset I$ so $cd \in I.$ Therefore $I = (cd)$ is principal, contradiction. $\ \ $ QED
I claim that the second part of the proof is just an ideal theoretic
version of a well-known fact about integers. Namely suppose that the
integer $i>1$ isn’t prime. Then, by definition, there are integers $a,b$
such that $i|ab,\ \,i\nmid a,b.$ But this immediately yields a
proper factorization of $i,$ namely $i = c\, (i\!:\!c),$ where $c = (i,a).$
Therefore: not prime $\Rightarrow$ reducible (or: irreducible $\Rightarrow$ prime).
A similar constructive proof works much more generally, namely
Theorem $ $ If ideal $I\ne 1$ satisfies: ideal $\,J \supset I \Rightarrow J\,|\,I\,$ then $I$ not prime $\Rightarrow I\,$ reducible (properly).
Proof $\ $ $I$ not prime $\Rightarrow$ exists $a,b \not\in I$ and $ab \in I.$
$\ A := (I,a)\supset I \Rightarrow A|I,$ say $I = AB;$ wlog we may assume $b \in B$
since $A(B,b) = AB\,$ via $Ab = (I,a)b \subset I = AB.$ The factors $A,B$
are proper: $A = (I,a),\, a \not\in I;\,\ B \supset (I,b),\, b \not\in I.\quad$ QED
Note that the contains $\Rightarrow$ divides hypothesis: $J\supset I \Rightarrow J\,|\,I\,$
is trivially true for principal ideals $J$ (hence proof$_{\,1}$),
and also holds true for all ideals in a Dedekind domain.
Generally such ideals J are called multiplication ideals.
Rings whose ideals satisfy this property are known as
multiplication rings and their study goes back to Krull.
The OP’s problem is well-known: it is Exercise $1\!-\!1\!-\!10\ p.8$
in Kaplansky: Commutative Rings, namely:
For generalizations of such Kaplansky-style Zorn Lemma arguments see the papers referenced in my post here.
Below is an interesting reference on multiplication rings.
Mott, Joe Leonard. Equivalent conditions for a ring to be a multiplication ring.
Canad. J. Math. 16 1964 429–434. MR 29:119 13.20 (16.00)
If “ring” is taken to mean a commutative ring with identity and a
multiplication ring is a “ring” in which, when A and B are ideals with
A $\subset$ B, there is an ideal C such that A = BC , then it is shown that
the following statements are equivalent.
Reviewed by H. T. Muhly
One can prove this using the machinery from Lam and Reyes’s Prime ideal principle (available here). A beautiful paper that I highly recommend to anyone interested in this kind of arguments.
Given an ideal $I$ and an element $a\in R$, define $I:a = \{x\in R\mid xa\in I\}$. Note that this is an ideal.
Definition. A class of ideals $\mathcal{F}$ of $R$ with $R\in\mathcal{F}$ is an Oka family if for $a\in R$ and $I\triangleleft R$, if both $(I,a)$ and $I\colon a$ are in $\mathcal{F}$, then $I$ is in $\mathcal{F}$.
Definition. Let $\mathcal{F}$ be a class of ideals of $R$. We say that $\mathcal{F}$ is an MP-family (“Maximal is prime”) if the maximal elements of $\mathcal{F}$ are prime ideals of $R$.
Definition. Let $\mathcal{F}$ be a class of ideals of $R$. Then $\mathcal{F}’$ is the complement of $\mathcal{F}$ in the class of all ideals of $R$.
Theorem. (Lam and Reyes) If $\mathcal{F}$ is an Oka family, then $\mathcal{F}’$ is an MP-family.
Proof. Let $I$ be a maximal element of $\mathcal{F}’$ and assume that $I$ is not prime. Since $I\neq R$ (as $R\in\mathcal{F}$ by definition), let $a,b\in R$ such that $a,b\notin I$, such that $ab\in I$. Then $(I,a)$ is strictly larger than $I$, hence in $\mathcal{F}$; and since $ba\in I$, then $b\in I:a$, hence $I:a$ also properly contains $I$. Thus, $(I,a)$ and $(I:a)$ both lie in $\mathcal{F}$. Therefore, since $\mathcal{F}$ is an Oka family, it follows that $I\in \mathcal{F}$, contradicting that $I\in\mathcal{F}’$. $\Box$
Theorem. Let $R$ be a commutative ring with unity, and suppose that $R$ has at least one nonprincipal ideal. If $I$ is an ideal that is maximal with respect to the property of being nonprincipal (that is, if $I\subsetneq J$ and $J$ is an ideal, then $J$ is principal), then $I$ is a prime ideal.
Proof. We prove that the collection of principal ideals is an Oka family. It clearly contains $R$. Suppose $(I,a)$ and $I:a$ are principal. Let $(I,a) = (x)$ and $I:a = (y)$.
I claim that $I=(xy)$. Indeed, first note that $(xy)\subseteq I$: $(x)=I+(a)$ so $(xy)=(x)(y) = (I+(a))(y) = Iy + (ay)\subseteq I+I = I$, so $(xy)\subseteq I$. Conversely, if $r\in I$, then $r=xt$ for some $t$; since $a=xv$ for some $v$, we have $vr=vxt = at\in I$, so $t\in I:a=(y)$, hence $t=yz$ for some $z$, so $r=xt=xyz\in (xy)$. Thus, $I=(xy)$.
Therefore, the collection of principal ideals is an Oka family. In particular, the collection of nonprincipal ideals is an MP family. But since every prime ideal is principal, it follows that the collection of nonprincipal ideals is empty, so every ideal of $R$ is principal. $\Box$
We do this in a few steps. Let $R$ be our integral domain, and we are supposing that every prime ideal in $R$ is principal.
Let $S$ be the set of non-principal ideals of $R$, and suppose that $S$ is nonempty. Throw an inclusion partial order on $S$ (or note that it has one, but I like to be physical with my math sometimes). WRT this partial order, let $\{C_i\}$ be a chain in $S$, and let $C = \cup C_i$. Then we know that $C$ is an ideal.
Prove that $C$ is not principal:
Suppose that $C$ is principal, so that $C = \langle c \rangle$. Then $c \in C_j$ for some $j$, so that $C \subseteq C_j$. Then $C_j$ is principal, contradicting our hypothesis.
Appeal to Zorn to get an inclusion-maximal nonprincipal ideal:
As $C$ isn’t principal, the chain $\{C_i\}$ has an upper bound in $S$. By Zorn’s Lemma, $S$ contains a maximal element. Let $M$ be one such element.
Let $a,b \in R$ s.t. $ab \in M$ while $a,b \not \in M$, which exist because $M \in A$ and thus is not prime. Since $a \not \in M$, we have that $M \subsetneq (M,a)$, $M \subsetneq (M,b)$. As $M$ is maximal in terms of non-principality, we know that both $(M,a)$ and $(M,b)$ are principal. Suppose that $(M,a) = \langle \alpha \rangle, (M,b) = \langle \beta \rangle$.
Let $N = \{ r \in R\;|\;r(M,a) \subseteq M\}$. Note that $(M,a)(M,b) = (M^2, Ma, Mb, ab) \subseteq M$, so that $(M,b) \subseteq N$. And further, $(M,a)N \subseteq M$, and $(M,a)N$ is principal.
If $x \in M$, we’ll show that $x = s\alpha$ for some $s \in N$, and conclude that $M = (M,a)N$ is principal – a contraxiction:
Let $x \in M$. Since $M \subset (M,a)$, we know that $x = s\alpha$ for some $s \in R$. Note that in fact $s(M,a) = s\langle \alpha \rangle \subseteq M$, so $s \in N$. Thus $M \subseteq (M,a)N$, and more importantly $M$ is principal.
This is a contradiction on the non-principality of $M$, so we are wrong to assume that $S$ is nonempty.
Thus $R$ has no nonprincipal ideals and is a principal ideal domain.