Abstract
A commutative ring R is said to have the two-generator property if each ideal of R can be generated by two elements, and is said to be stable if each regular ideal of R is projective over its endomorphism ring. It is known that for a one-dimensional local Macaulay ring the two-generator property implies stability but not conversely. We extend some of the known results on rings with the two-generator property to stable rings, and determine conditions under which stable rings have the two-generator property. We also extend a structure theorem for certain finitely generated torsionfree modules over rings with the two-generator property, and some of its consequences for cancellation of direct summands, to one-dimensional rings which may not have finite integral closure, and remove the finite integral closure hypothesis from a characterization of Greither of commutative group rings with the two-generator property.
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