  This paper investigates the local and global well-posedness for the
inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation $iu_{t} +\Delta
u=\lambda \left|x\right|^{-b} \left|u\right|^{\sigma } u, u(0)=u_{0} \in
L^{2}(\mathbb R^{n})$, where $\lambda \in \mathbb C$, $0<b<\min \left\{2,{\rm
\; }n\right\}$ and $0<\sigma \le \frac{4-2b}{n} $. We prove the local
well-posedness and small data global well-posedness of the INLS equation in the
mass-critical case $\sigma =\frac{4-2b}{n} $, which have remained open until
now. We also obtain some local well-posedness results in the mass-subcritical
case $\sigma <\frac{4-2b}{n} $. In order to obtain the results above, we
establish the Strichartz estimates in Lorentz spaces and use the contraction
mapping principle based on Strichartz estimates.
