When is $M \times N$ contained in$ M \otimes_{R} N$. Let $R$ be a ring with 1. $M$ is a right R-module and $N$ is a left R-module. A tensor product of M and N comes with a map from $M \times N \to M\otimes N$ which is actually a composition of maps obtained from $M \times N \to F_{\mathbb{Z}}(M \times N) \to M\otimes N$ where $F_{\mathbb{Z}}(M \times N)$ is free abelian group over $M\times N$. the first map is an injection while the second is a surjection. My question is
Is there a condition when we can state that this composition of maps is an injection?

Only when $M=N=0$. Look at $0 \otimes n = 0 = m \otimes 0$.
