Let $(R,\mathfrak{ m})$ be a $d$-dimensional Cohen-Macaulay local ring, $I$ an $\mathfrak{ m}$-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 $e_2(I) = \sum_{ n=2}^\infty (n-1) \lambda (I^n/JI^{ n-1})-2;$ or if $I$ is integrally closed and $e_2(I) = \sum_{ n=2}^\infty (n-1)\lambda({ { I^{ n}}}/JI^{ n-1})-i$ where $i=3,4$, then $e_1(I) = \sum_{ n=1}^\infty \lambda(I^n / JI^{ n-1})-1.$ In addition, we show that $r(I)$ is independent. Furthermore, we study the independence of $r(I)$ with some other conditions.
Categories: math.AC