Every scheme here is over complex number.

Let $X \subset (\mathbb{C}^*)^n$ be a complete intersection with $X$ defined by the ideal $I \subset \mathbb{C}[x_{1}^{\pm},\dots,x_{n}^{\pm}]$ generated by $r$ equations $f_1,\dots,f_r$. Moreover, if the rank of the matrix $(\partial f_i /\partial x_j)_{ij}$ is $r$, then is $X$ a smooth variety?

This is of course true if $I$ is radical, that is, $X$ is a variety. But I don't know any information about $I$. Actually, in my problem, I just want to know whether $I$ is reduced under the condition that the generators satisfying the Jacobian matrix smooth criterion(i.e. rank$(\partial f_i /\partial x_j)_{ij}=r$).