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چکیده
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Let $(A,\fm)$ be a commutative quasi-local ring with non-zero identity and $M$ be an Artinian $A$-module with $\dim M=d$. If $I$ is an ideal of $A$ with $\ell(0:_MI)<\infty$, then we show that for a minimal reduction $J$ of $I$, $(0:_{M}{J}I)=(0:_{M}I^2)$ if and only if $\ell(0:_{M}I^{n+1})=\ell(0:_{M}{J})\binom{n+d}{d}-\ell(\frac{0:_{M}{J}}{0:_{M}I})\binom{n+d-1}{d-1},$ for all $n\geq 0$. Also we study the dual of Burch's inequality. In particular, the Burch's inequality is equality if $G(I,M)$ is co-Cohen-Macaulay.
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