Abstract
Let (R,m) be a Noetherian local ring of dimension d ≥ 2. We prove that if e(Rred) > 1, then the classical Lech’s inequality can be improved uniformly for all m-primary ideals, that is, there exists ε > 0 such that e(I) ≤ d!(e(R) − ε)l(R/I) for all m-primary ideals I ⊆ R. This answers a question raised in [3]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of I.