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