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 On the first generalized Hilbert coefficient and depth of associated graded rings Bull. Korean Math. Soc.Published online October 24, 2019 Amir Mafi and Dler Naderi University of Kurdistan, P.O. Box: 416, Sanan- daj, Iran. Abstract : ‎Let $(R,\frak{m})$ be a $d$-dimensional Cohen-Macaulay local ring with infinite residue field‎. ‎Let $I$ be an ideal of $R$ that has analytic spread $\ell(I)=d$‎, ‎satisfies the $G_d$ condition‎, ‎the weak Artin-Nagata property $AN_{d-2}^-$ and $\frak{m}$ is not an associated prime of $R/I$‎. ‎In this paper‎, ‎we show that if $j_1(I) = \lambda (I/J)‎ +‎\lambda [R/(J_{d-1}‎ :‎_{R} I+(J_{d-2}‎ :‎_{R}I+I)‎ ‎:_{R}{\frak{m}}^{\infty})]+1$,‎ ‎then $I$ has almost minimal $j$-multiplicity‎, ‎$G(I)$ is Cohen-Macaulay and $r_J(I)$ is at most 2‎, ‎where $J=(x_1‎, . . . ,‎x_d)$ is a general minimal reduction of $I$ and $J_i=(x_1‎, . . . ‎,x_i)$.‎ ‎In addition‎, ‎the last theorem is in the spirit of a result of Sally who has studied the depth of associated graded rings and minimal reductions for an $\frak{m}$-primary ideals‎. Keywords : Generalized Hilbert coefficient‎, ‎Minimal reduction‎, ‎Associated graded ring MSC numbers : 13A30‎, ‎13D40‎, ‎13H15,13C14 Full-Text :