The zero tensor-divisors of 𝕜-algebras

Let $A$ be a 𝕜-algebra defined over a field 𝕜. This paper consider the two questions: If the tensor $M{\otimes }_{A}N$ on $A$ is zero, under what situation is either $M=0$ or $N=0$? If the element, say $m\otimes n$, in $M{\otimes }_{A}N$ is zero, under what situation is either $m=0$ or $n=0$? This study introduces strong/weak zero-tensor-divisors in this paper and provide a solution of the above questions by quiver methods. To be precise, assume that $A$ is a connected algebra whose quiver contains at most one loop, then have the results as follows: (1) $A$ has strong zero-tensor-divisors, the number of vertices of its quiver is greater than or equal to 2, the quiver of $A$ is not a loop. (2) If $A$ does not have weak zero-tensor-divisors then its dimension is infinite.