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You may find Matlis' paper The Two-Generator Problem for Ideals to be interesting, as its main theorem concerns the class of integral domains in which every ideal is generated by two elements.

It was proven by Cohen (in Commutative Rings with Restricted Minimum Condition ) that an integral domain with the property that there exists an integer $n$ such that every ideal can be generated with fewer than $n$ elements must be Noetherian and of Krull dimension 1.

Say that an integral domain R has property FD if every finitely generated torsion free R-module is direct sum of modules of rank 1. Moreover, say that R has property FD locally if RM has property FD for every maximal ideal M of R.

Theorem (simplified form) - Let R be an arbitrary integral domain. Then every ideal of R can be generated by two elements if and only if R is a noetherian ring that has property FD locally.

 - Here is an example of an integral domain in which not every ideal can be two-generated. Let $R$ be any Dedekind domain and consider the polynomial ring $R[x]$. It has Krull dimension at least 2, so by Cohen's theorem (mentioned above), there exist ideals of $R[x]$ requiring arbitrarily many generators.

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You may find Matlis' paper The Two-Generator Problem for Ideals to be interesting, as its main theorem concerns the class of integral domains in which every ideal is generated by two elements.

It was proven by Cohen (in Commutative Rings with Restricted Minimum Condition ) that an integral domain with the property that there exists an integer $n$ such that every ideal can be generated with fewer than $n$ elements must be Noetherian and of Krull dimension 1.

Say that an integral domain R has property FD if every finitely generated torsion free R-module is direct sum of modules of rank 1. Moreover, say that R has property FD locally if RM has property FD for every maximal ideal M of R.

Theorem (simplified form) - Let R be an arbitrary integral domain. Then every ideal of R can be generated by two elements if and only if R is a noetherian ring that has property FD locally.

 - Here is an example of an integral domain in which not every ideal can be two-generated. Let $R$ be any Dedekind domain and consider the polynomial ring $R[x]$. It has Krull dimension at least 2, so by Cohen's theorem (mentioned above), there exist ideals of $R[x]$ requiring arbitrarily many generators.

2 added 251 characters in body

You may find Matlis' paper The Two-Generator Problem for Ideals to be interesting, as its main theorem concerns the class of integral domains in which every ideal is generated by two elements.

It was proven by Cohen (in Commutative Rings with Restricted Minimum Condition ) that an integral domain with the property that there exists an integer $n$ such that every ideal can be generated with fewer than $n$ elements must be Noetherian and of Krull dimension 1.

Say that an integral domain R has property FD if every finitely generated torsion free R-module is direct sum of modules of rank 1. Moreover, say that R has property FD locally if RM has property FD for every maximal ideal M of R.

Theorem (simplified form) - Let R be an arbitrary integral domain. Then every ideal of R can be generated by two elements if and only if R is a noetherian ring that has property FD locally(and has Krull dimension 1).

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