Let \(R\) be a commutative ring and let \(M\) and \(N\) be \(R\)-modules. The \(R\)-module structure on \(M\tensor_R N\) is the scalar multiplication determined on pure tensors by \[ r\cdot (m\tensor n) \coloneqq (rm)\tensor n, \qquad r\in R,\ m\in M,\ n\in N. \]
The scalar multiplication is well defined: for fixed \(r\in R\), the map \(M\times N\to M\tensor_R N\), \((m,n)\mapsto (rm)\tensor n\), is \(R\)-bilinear, so it induces an additive map \(M\tensor_R N\to M\tensor_R N\). Since \((rm)\tensor n = m\tensor (rn)\), the same structure is determined by \(r\cdot(m\tensor n) = m\tensor(rn)\).