The monoidal center construction and bimodules

Let E be a cocomplete monoidal category such that the tenser product in E preserves colimits in each argument. Let A be an algebra in E. We show (under some assumptions including faithful flatness of A) that the center of the monoidal category ((A)E(A), circle times (A)) of A-A-bimodules is equivalent to the center of E (hence in a sense trivial): L((A)E(A)) congruent to L(E) Assuming A to be a commutative algebra in the center L(E), we compute the center L(E(A)) of the category of right A-modules (considered as a subcategory of (A)E(A) using the structure of A is an element of L(E). We find L(E(A)) congruent to dys L(E)(A), the category of dyslectic right A-modules in the braided category L(E) (C) 2001 Elsevier Science B.V. All rights reserved.