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چکیده
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Let $(R,\fm)$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\fm$-primary ideal of $R$ and $J=(x_1,...,x_d)$ a minimal reduction of $I$. We show that if $J_{d-1}=(x_1,...,x_{d-1})$ and $\sum\limits_{n=1}^\infty\lambda{({{I^{n+1}}\cap J_{d-1}})/({J{I^n} \cap J_{d-1}})=i}$ where i=0,1, then $\depth G(I)\geq{d-i-1}$. Moreover, we prove that if ${e_2}(I)=\sum\limits_{n = 2}^\infty{(n-1)\lambda({{I^{n }}}/{J{I^{n-1}}})}-2;$ or if $I$ is integrally closed and\\ ${e_2}(I)=\sum\limits_{n=2}^\infty{(n-1)\lambda({{I^{n}}}/{J{I^{n-1}}})}-3$, then ${e_1}(I)=\sum\limits_{n=1}^\infty{\lambda({{I^{n }}}/{J{I^{n-1}}})}-1,$ where the integers $e_i$ are the Hilbert coefficients of $I$. In addition, if $J$ is a minimal reduction of $I$ then we prove that the reduction number $r_J(I)$ is independent of $J$.
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