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I am trying to find a right $R$-module $A$ and a left $R$-module $B$ such that the tensor product $A\otimes_R B$ has an element that is not of the form $a\otimes b$ for any $a,b$ in $A,B$ respectively. I could not find any. Is there any simple example? Thanks in advance
Edit: You may bring other examples, or consider my own example that taking $A,B$ to be $K^2$ for a field $K$, then $e_1\otimes e_1 + e_2\otimes e_2$ where $e_i$ are the canonical basis elements of $K$ is such an element but how can it be proved using elementary methods?