A Generalized Principal Ideal Theorem

KrulΓs principal ideal theorm [Krull] states that q elements in the maximal ideal of a local noetherian ring generate an ideal whose minimal components are all of height at most q. Writing R for the ring, we may consider the q elements, x19 , xq say, as coordinates of an element xeR. It is an easy observation that every homomorphism R —> R carries x to an element of the ideal generated by xi9 ,xq. Write R * for HonΓβ (Rq9 R) and set jββ (a) = [ψ{χ) i φ e Horn (R q 9 R)}