Checking if an element of tensor product is zero I am trying to understand tensor product.

Let $R$ be a ring and $R_0$ its subring. Then $R$ is a right $R_0$-module. Let $M$ be a left $R_0$-module.


  Is the element $1\otimes m$ of $R\otimes_{R_0} M$ nonzero for a nonzero $m\in M$?


I tried to express $(1,m)$ of $R\times M$ as a linear combination of elements of the form 
$(r_1+r_2,m)-(r_1,m)-(r_2,m)$,
$(r,m_1+m_2)-(r,m_1)-(r,m_2)$,
$(rr_0,m)-(r,r_0m)$
or to see that such an expression is not possible, but failed at both.

I know how to prove this in the affirmative for vector spaces (if $R_0$ is a field) by choosing a basis, and I also know that in some tensor products $1\otimes m$ can be zero, e.g. in  $\mathbb Z_m\otimes \mathbb Z_n$ for $(m,n)=1$, but it does not help me to solve this problem.
Not necessarily! The tensor product $\mathbb Q\otimes_{\mathbb Z}\mathbb Z/n\mathbb Z$ is zero.
The element $1\otimes m$ is $0$ precisely if $m$ is annihilated by an element of $R_0$. Thus the answer to your question is yes if and only if $\operatorname{Ann}_{R_0}(M)\not=0$.