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Let R be a commutative ring with identity. If every ideal generated by two elements of R is principal, then can we conclude that R is a PID? Also, if every finitely generated ideal of R is principal, can we conclude that R is a PID?

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The first option doesn't work since in finite algebraic extension of $\,\Bbb Q\,$ , we know that the fractional ideals (of the Dedekind domain of integers of the extension) can always be generated by two elements, and thus the actual (integral) ideals too, but the ring can be far from principal. About the second option I can't tell right now, but what we know is that if every prime ideal is principal then the ring is principal. This is a non-trivial claim and if I remember correctly we need Zorn's Lemma to prove it (the set of all non-principal ideals and etc.) –  DonAntonio Oct 22 '12 at 4:57
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Dear @DonAntonio: I don't think your example works, because the OP assumes that in his ring, every ideal generated by 2 elements is principal. What you are saying is that there are rings where every ideal is generated by at most two elements but where the rings are not principal. Certainly this does not violate what the OP is asking. –  Rankeya Oct 22 '12 at 5:00
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Dear user, Note that your two conditions (every ideal gen. by $2$ elts. is principal, and every f.g. ideal is principal) are equivalent (by an induction on the number of generators). Regards, –  Matt E Oct 22 '12 at 5:20
    
Dear @Rankeya, you of course are right. I misread. Thanks. –  DonAntonio Oct 22 '12 at 10:48

2 Answers 2

up vote 4 down vote accepted

A domain with every two-generated ideal principal $\rm\:(a,b) = (c)\:$ is called a Bezout domain. By induction this is equivalent to every finitely-generated ideal being principal. This does not imply that every ideal is principal (PID). Indeed, the ring of all algebraic integers is Bezout (Kaplansky, Commutative Rings, Theorem 102), but its ideal $\:(2^{1/2},\,2^{1/3},\ldots)\:$ is not principal.

A domain is a PID iff it's Bezout and satisfies ACCP (ascending chain condition on principal ideals). The direction $(\Rightarrow)$ is clear. For $(\Leftarrow)$ note that ACCP implies that the divisibility relation is well-founded, hence a nonzero ideal I is generated by an element c minimal wrt to divisibility: for if $\rm\:c\nmid i\in I\:$ then $\rm\:(c,i) = (d)\subset I\:$ and $\rm\:d\:|\:c\:$ properly, contra minimality of c. This is essentially an application of the Dedekind-Hasse Test for a PID.

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No, in general this is not true. For an example, in the ring of entire functions every finitely generated ideal is principal, but this ring is not a PID.

If, in addition, $R$ satisfies the ascending chain condition for principal ideals, then one can conclude that $R$ is a PID.

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