Perfect pairing

Let \(R\) be a commutative ring and \(M\), \(N\), \(L\) \(R\)-modules. A pairing of \(M\) and \(N\) with values in \(L\) is an \(R\)-bilinear map \(p\colon M\times N\to L\), equivalently an \(R\)-linear map \(M\otimes_R N\to L\) on the tensor product. The pairing \(p\) is perfect if the \(R\)-linear map \[ \phi\colon M\to\Hom_R(N, L),\qquad \phi(m)(n)\coloneqq p(m, n), \] is an isomorphism.