Chains of Modules, A Prime in a Principal Ideal

A Prime in a Principal Ideal

Let R be a commutative noetherian ring. Let g be a nonunit that generates a principal ideal, and let P be a prime ideal that is properly contained in g*R. For every y in P, y = gx for some x. Since P is prime, x belongs to P. Thus P is contained in gP, and of course, gP is contained in P, hence P = gP.

Let H₀ be an ideal that lives inside P. Let H₁ be the conductor ideal that drives g into H₀. This is sometimes denoted [H₀:g]. Of course H₁ includes all of H₀, and since g generates H₀, g*H₁ = H₀. Since g lies outside of P, H₁ lies in P. Let H₂ = [H₁:g], let H₃ = [H₂:g], and so on.

The ascending chain cannot continue forever, so say H₆ = H₇. Now gH₇ = H₆ and gH₆ = H₅. This forces H₆ = H₅, even though H₆ is larger than H₅. The only way around this contradiction is to assume H₀ = H₁. In other words, gH₀ = H₀. Multiplication by g maps each ideal in P onto itself.

Next assume R is a noetherian integral domain. Select g and P as above. Let z generate a principal ideal inside P. Thus gz generates z, or gxz = z, making g a unit. This is a contradiction, hence there is no prime ideal strictly between 0 and g*R.