Modules

• How to Prove an Element Is Nonzero in a Tensor Product
  Over a field, detect a nonzero tensor by applying linear functionals; over a ring, use a balanced map or compute the module relations.
• Is the Tensor Product of a Module Element and Zero Always Zero?
  In a tensor product of modules, m ⊗ 0 and 0 ⊗ n are always zero. Bilinearity proves this, but sums of pure tensors need separate analysis.
• How to Test Whether an Element of a Tensor Product Is Zero
  To test whether a tensor is zero, use a module presentation or a separating map; pure tensors and vector-space tensors have useful special criteria.
• When Is an Element of a Tensor Product Zero?
  For vector spaces over a field, a pure tensor is zero exactly when one factor is zero; sums and tensors over general rings need separate tests.
• When Is the Tensor Product Commutative?
  Tensor products of modules over a commutative ring are symmetric up to a natural flip isomorphism, but the factors are not literally interchangeable.