Tensor product of modules over a vertex algebra

We find a necessary and sufficient condition for the existence of the tensor product of modules over a vertex algebra. We define the notion of vertex bilinear map and provide two algebraic constructions of the tensor product, where one of them is of ring theoretical type. We show the relation between tensor product and vertex homomorphisms. We prove commutativity of the tensor product. We also prove associativity of the tensor product of modules under certain necessary and sufficient condition. Finally, we show certain functorial properties of vertex homomorphism and the tensor product.