$(R,m)$ is local neotherian cohen-macaulay ring of dimension $d$, and $I$ is an $m$-primary ideal of $R$. since $I$ is an $m$-primary, $\dim R /I=\dim R/I^n =0$. so $\ell(R/I^n)$ and $\ell (R/I)$ are finite.
i saw in papers, when $d=1$, they say $\ell (R/I^n)=n.\ell (R/I)$. but I dont know why. and I wonder if this can be generalized to arbitrary dimension? so the question is:
is there a relationship between $\ell (R/I^n)$ and $\ell (R/I)$?
thank you