  Let $(R,\mathfrak{m})$ 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 depth$(R/I)\geq\min\lbrace 1,\dim R/I \rbrace$. In this paper,
we show that if $j_1(I) = \lambda (I/J) +\lambda [R/(J_{d-1} :_{R} I+(J_{d-2}
:_{R}I+I) :_R, \mathfrak{m}^\infty)]+1$, then depth$(G(I))\geq d -1$ and
$r_J(I)\leq 2$, where $J$ is a general minimal reduction of $I$. In addition,
we extend the result by Sally who has studied the depth of associated graded
rings and minimal reductions for an $,\mathfrak{m}$-primary ideals.

Commutative Algebra;Algebraic Geometry