(tensor product)
Let \(B \to A\) be a map of rings and \(M\) a \(B\)-module. Any \(A\)-module is naturally a \(B\)-module; to endow \(M\) with an \(A\)-module structure one forms \(M \tensor_B A\), extension of scalars, and \(\wait \tensor_B A\) is a covariant functor from \(B\)-modules to \(A\)-modules.
For \(A\)-modules \(M\) and \(N\), the tensor product \(M \tensor_A N\) satisfies a universal property: for any \(A\)-bilinear map \(M \times N \to T\) there is a unique map of \(A\)-modules \(M \tensor_A N \to T\) through which it factors.
It exists by explicit construction, as the \(A\)-linear combinations of simple tensors \(m \tensor n\) subject to \[ \begin{aligned} (a_1m_1 + a_2m_2)\tensor n &= a_1(m_1 \tensor n) + a_2(m_2\tensor n), \\ m \tensor (b_1n_1 + b_2n_2) &= b_1(m\tensor n_1) + b_2(m \tensor n_2), \\ a(m\tensor n) &= (am)\tensor n = m \tensor (an) , \end{aligned} \] with the bilinear map \((m,n)\mapsto m \tensor n\).
The functor \(\wait\tensor_AN\) is left adjoint to \(\Hom_A(N,\wait)\), so it preserves colimits, in particular cokernels, and is right exact (D-DEFADJ). For a short exact sequence \(0\to M'\to M\to M''\to0\), the sequence \[ \Tor_1^A(M'',N)\to M'\tensor_AN\to M\tensor_AN\to M''\tensor_AN\to0 \] is exact; for \(A=\ZZ\), \(N=\ZZ/2\), and \(0\to\ZZ\xrightarrow{2}\ZZ\to\ZZ/2\to0\), the map \(\ZZ\tensor\ZZ/2\to\ZZ\tensor\ZZ/2\) is zero, with kernel the image of \(\Tor_1^\ZZ(\ZZ/2,\ZZ/2)\cong\ZZ/2\).
A simple tensor \(m\tensor n\) can vanish with \(m\ne0\) and \(n\ne0\): in \(\ZZ \tensor_\ZZ \ZZ/2\), \(2\tensor1=1\tensor2=0\). Not every element is a simple tensor: \(e_1\tensor e_1+e_2\tensor e_2\in k^2\tensor_kk^2\) corresponds to the identity matrix, which has rank \(2\), while simple tensors correspond to matrices of rank at most \(1\). For ring maps \(C\to A\) and \(C\to B\), \(\Spec A \times_{\Spec C} \Spec B = \Spec(A \tensor_C B)\).