pure tensor
A pure tensor is the simplest kind of tensor: a single product of vectors, like v ⊗ w, with nothing added together. These are the 'atoms' from which every tensor is assembled by addition — but, importantly, not every tensor is itself an atom. The general tensor is a sum of pure tensors that often cannot be collapsed back to one.
In M ⊗ N, a pure tensor (also called simple or decomposable) is an element of the form m ⊗ n. Pure tensors span the whole tensor product, so every element is a finite sum Σ m_i ⊗ n_i, but this representation is far from unique and the sum is usually not itself a pure tensor. The pure tensors form the image of the universal bilinear map M × N -> M ⊗ N — a 'cone' that is not closed under addition.
Whether a tensor is pure is a real and computable question. Writing a tensor in V ⊗ W as a matrix of coefficients (with respect to bases), it is pure exactly when that matrix has rank ≤ 1; the minimal number of pure tensors needed in a sum is the tensor rank. In V ⊗ V, for instance, the symmetric element e_1 ⊗ e_2 + e_2 ⊗ e_1 is not pure — it has rank 2.
In R^2 ⊗ R^2, the tensor e_1 ⊗ e_1 + e_2 ⊗ e_2 corresponds to the coefficient matrix [1, 0; 0, 1], which has rank 2, so it is not a pure tensor; no single product v ⊗ w equals it. By contrast e_1 ⊗ e_1 + e_1 ⊗ e_2 = e_1 ⊗ (e_1 + e_2) is pure.
Rank-1 coefficient matrix means pure; the identity matrix means rank-2, not pure.