2-GENERATED IDEALS AND REPRESENTATIONS OF ABELIAN-GROUPS OVER VALUATION RINGS

Let R be a commutative ring with identity. We give some general result s on non-Noetherian commutative rings with the property that each fini tely generated ideal can be generated by n elements, and characterize the quasi-local reduced group rings R[G], and closely related group ri ngs 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. Th e group rings R[G] which have only finitely many isomorphism classes o f finitely generated torsionfree modules are also determined, where th e coefficient ring R is as in the above-mentioned characterization of when R[G] has the 2-generator property. These results depend on a dete rmination of when a simple ring extension of the form R[X]/(Phi(pr)(X) ) is a valuation domain for a prime power p(r), where Phi(pi)(X) = X(p i-1(p-1)) + ... +X(p i-1) + 1, and some related results, which are giv en in Section 3. The relationship between the 2-generator property and stability of finitely generated regular ideals is also considered. (C ) 1995 Academic Press, Inc.