# ON THE FIRST GENERALIZED HILBERT COEFFICIENT AND DEPTH OF ASSOCIATED GRADED RINGS

• Mafi, Amir (Department of Mathematics University Of Kurdistan) ;
• Naderi, Dler (Department of Mathematics University Of Kurdistan)
• Accepted : 2019.09.19
• Published : 2020.03.31

#### Abstract

Let (R, m) be a d-dimensional Cohen-Macaulay local ring with infinite residue field. Let I be an ideal of R that has analytic spread ℓ(I) = d, satisfies the Gd condition, the weak Artin-Nagata property AN-d-2 and m is not an associated prime of R/I. In this paper, we show that if j1(I) = λ(I/J) + λ[R/(Jd-1 :RI+(Jd-2 :RI+I):R m)] + 1, then I has almost minimal j-multiplicity, G(I) is Cohen-Macaulay and rJ(I) is at most 2, where J = (x1, , xd) is a general minimal reduction of I and Ji = (x1, , xi). 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 m-primary ideals.

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