The aim of this paper is to introduce a tensor structure for the Serre quotient category of an abelian monoidal category with biexact tensor product to make the canonical functor a monoidal functor. In this tensor product, the Serre quotient category of a multiring category (resp. a multitensor category) by a two-sided Serre tensor-ideal is still a multiring category (resp. a multitensor category). Besides, a two-sided Serre tensor-ideal of a tensor category is always trivial. This result can be generalized to any tensor product. If the canonical functor is a monoidal functor, then the corresponding Serre subcategory of the tensor category is trivial.
