  In this paper, we study tensor (or monoidal) categories of finite rank over
an algebraically closed field $\mathbb F$. Given a tensor category
$\mathcal{C}$, we have two structure invariants of $\mathcal{C}$: the Green
ring (or the representation ring) $r(\mathcal{C})$ and the Auslander algebra
$A(\mathcal{C})$ of $\mathcal{C}$. We show that a Krull-Schmit abelian tensor
category $\mathcal{C}$ of finite rank is uniquely determined (up to tensor
equivalences) by its two structure invariants and the associated associator
system of $\mathcal{C}$. In fact, we can reconstruct the tensor category
$\mathcal{C}$ from its two invarinats and the associator system. More general,
given a quadruple $(R, A, \phi, a)$ satisfying certain conditions, where $R$ is
a $\mathbb{Z}_+$-ring of rank $n$, $A$ is a finite dimensional $\mathbb
F$-algebra with a complete set of $n$ primitive orthogonal idempotents, $\phi$
is an algebra map from $A\otimes_{\mathbb F}A$ to an algebra $M(R, A, n)$
constructed from $A$ and $R$, and $a=\{a_{i,j,l}|1< i,j,l<n\}$ is a family of
"invertible" matrices over $A$, we can construct a Krull-Schmidt and abelian
tensor category $\mathcal C$ over $\mathbb{F}$ such that $R$ is the Green ring
of $\mathcal C$ and $A$ is the Auslander algebra of $\mathcal C$. In this case,
$\mathcal C$ has finitely many indecomposable objects (up to isomorphisms) and
finite dimensional Hom-spaces. Moreover, we will give a necessary and
sufficient condition for such two tensor categories to be tensor equivalent.
