When we write tensorial tensor products, its it's optional to indicate the ring over which we do it; we can write M⊗N ⊗ N or M ⊗R N. But for elements, we always type write x⊗y ⊗ y without reference to R. You must rest keep it in mind and that can induces lapsusinduce lapses. For example, v⊗u ⊗ u2- − u⊗uv maybe ⊗ uv may be ≠0: It it depends of on the base ring, and it don´t that doesn't appear [in the notation].
Sometimes the problem it is not the concept like so much as the notation we use for it.
