Kronecker pairing descends to homology and cohomology

Let \(R\) be a ring. If \[ z=\sum_i r_i\sigma_i \] is a \(k\)-cycle in \(C_k(X;R)\) and \(\alpha\in C^k(X;R)\), define \[ \langle z,\alpha\rangle=\sum_i r_i\,\alpha(\sigma_i). \] Prove that if \(z-z'\) is a boundary, then \(\langle z,\alpha\rangle=\langle z',\alpha\rangle\). Likewise, prove that if \(\alpha\) is a coboundary and \(z\) is a cycle, then \(\langle z,\alpha\rangle=0\). Deduce that there is a well-defined bilinear pairing \[ H_k(X;R)\times H^k(X;R)\longrightarrow R. \] This pairing is sometimes called the Kronecker product.