On ideals of rings of fractions and rings of polynomials

Abstract

We investigate the links between the lattice Idl( R) of ideals of a commutative ring R and the lattices Idl( R′) of ideals of various new rings R′ constructed from R, in particular, the ring S ⁻¹ R of fractions and the ring R[ X] of polynomials. For any partially ordered set P, we construct another poset N( P) and show that P satisfies the ascending chain condition if and only if N( P) satisfies the ascending chain condition. As an application of this result, we give an order version proof for Hilbert's Basis Theorem.