Two-Generated Ideals and Representations of Abelian Groups over Valuation Rings

Abstract

Let R be a commutative ring with identity. We give some general results on non-Noetherian commutative rings with the property that each finitely generated ideal can be generated by n elements, and characterize the quasi-local reduced group rings R[ G], and closely related group rings which have this property for n = 2. It is then shown that finitely generated torsionfree R[ G]-modules are direct sums of ideals if R[ G] is a reduced quasi-local group ring with this 2-generator property. The group rings R[ G] which have only finitely many isomorphism classes of finitely generated torsionfree modules are also determined, where the coefficient ring R is as in the above-mentioned characterization of when R[ G] has the 2-generator property. These results depend on a determination of when a simple ring extension of the form R[ X]/(Φ p r ( X)) is a valuation domain for a prime power p r , where Φ p i ( X) = X p i−1 ( p−1) + ··· + X p i−1 + 1, and some related results, which are given in Section 3. The relationship between the 2-generator property and stability of finitely generated regular ideals is also considered.