A bound on the reduction number of a primary ideal

Let $( A,{\cal M})$ be a local ring of positive dimension $d$ and let $I$ be an $\cal M$-primary ideal. We denote the reduction number of $I$ by $r(I)$, which is the smallest integer $r$ such that $I^{r+1}=JI^r$ for some reduction $J$ of $I.$ In this paper we give an upper bound on $r(I)$ in terms of numerical invariants which are related with the Hilbert coefficients of $I$ when $A$ is Cohen-Macaulay. If $d=1$, it is known that $r(I) \le e(I) -1$ where $e(I)$ denotes the multiplicity of $I.$ If $d \le 2,$ in Corollary 1.5 we prove $r(I) \le e_1(I) - e(I) + \lambda (A/I) + 1$ where $e_1(I)$ is the first Hilbert coefficient of $I.$ From this bound several results follow. Theorem 1.3 gives an upper bound on $r(I)$ in a more general setting.