Projective Modules, Nonzero Tensor Product

Nonzero Tensor Product

Let A×B = T, where T is nonzero. Select a nonzero element c of T. Now c is a linear combination of product pairs, e.g. x₁y₇+3x₅y₂-29x₄y₈. Let the aforementioned values of x generate a submodule in side A and let the values of y generate a submodule inside B. Tensor these two submodules and find the same element c. There are even fewer relations corresponding to bilinearity, so c remains nonzero. Thus the tensor product of these two finitely generated modules is nonzero.

If the tensor product of every finitely generated submodule of A with every finitely generated submodule of B is zero then A×B is zero. This follows directly from the above.

If A×B is zero, not much can be asserted about their submodules. For instance, let A = Z_p, the integers mod p, and let B = Q, the rationals. Both are Z modules. What does the tensor product look like? Recall that 0 in either component maps to 0. If the second component is a nonzero rational number it is divisible by p. Move p to the other side and make the first component zero. Therefore the tensor product is 0. However, if we restrict B to Z, a submodule of Q, Z_p tensor Z = Z_p, which is nonzero. Restricting attention to a finitely generated submodule can turn a zero tensor product into something nonzero.