  For a braided tensor category C and a subcategory K there is a notion of
centralizer C_C(K), which is a full tensor subcategory of C. A pre-modular
tensor category is known to be modular in the sense of Turaev iff the center
Z_2(C):=C_C(C) (not to be confused with the center Z_1 of a tensor category,
related to the quantum double) is trivial, i.e. equivalent to Vect, and
dim(C)<>0. Here dim(C)=sum_i d(X_i)^2, the X_i being the simple objects. We
prove the following double centralizer theorem: Let C be a modular category and
K a full tensor subcategory closed w.r.t. direct sums, subobjects and duals.
Then C_C(C_C(K))=K and dim(K)dim(C_C(K))=dim(C). We give several applications,
the most important being the following. If C is modular and K is a full modular
subcategory, then also L=C_C(K) is modular and C is equivalent as a ribbon
category to the direct product of K and L. Thus every modular category
factorizes (non-uniquely, in general) into prime ones. We study the prime
factorizations of the categories D(G)-Mod, where G is a finite abelian group.
