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   Let $(R,\mathfrak{m})$be a Noetherian local ring of dimension $d\ge <\#comment/>2$. We prove that if $e\left({\stackrel{^<\#comment/>}{R}}_{red}\right)>1$, then the classical Lech’s inequality can be improved uniformly for all $\mathfrak{m}$-primary ideals, that is, there exists $\mathrm{\epsilon <\#comment/>}>0$such that $e\left(I\right)\le <\#comment/>d!\left(e\right(R)-<\#comment/>\mathrm{\epsilon <\#comment/>})\mathrm{\ell <\#comment/>}\left(R/I\right)$for all $\mathfrak{m}$-primary ideals $I\subseteq <\#comment/>R$. This answers a question raised by Huneke, Ma, Quy, and Smirnov [Adv. Math. 372 (2020), pp. 107296, 33]. We also obtain partial results towards improvements of Lech’s inequality when we fix the number of generators of $I$. 

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   null (Ed.)