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In Smalo: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand:

Two orders are defined on the set of $d$-dimensional modules over a finite dimensional algebra $\Lambda$. One by $M\leq_{\operatorname{Hom}} N$ iff $\dim \operatorname{Hom}(X,M)\leq \dim \operatorname{Hom}(X,N)$ for all $X$ and one by $M\leq_n N$ iff $\dim \operatorname{Hom}(\Lambda^n/\Lambda^nA,M)\leq \dim \operatorname{Hom}(\Lambda^n/\Lambda^nA,N)$ for all $n\times n$-matrices $A$. It is now claimed that from Hilbert's basis theorem for $n$ large enough one gets that $\leq_n$ is equivalent to $\leq_{\operatorname{Hom}}$. Can somebody provide a more detailed argument?

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