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چکیده
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Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal of degree $d\leq 2$. We show that $(I^{k+1}:I)=I^k$ for all $k\geq 1$ and we disprove a motivation question that was asked by of Carlini, H\`a, Harbourne and Van Tuyl by providing of a counterexample. Also, by this counterexample, we give a negative answer to the question that the depth function of square-free monomial ideals are non-increasing.
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