When the tensor product of two module elements is nonzero

Question

What are the main cases in which we can say that $a \otimes b \neq 0 \in A \otimes B$, where $A$ and $B$ are $R$ modules?

It works for nonzero elements in free modules over an integral domain.

Additional Question: What can we say about when all tensors are elementary? It is true when one of the factors is cyclic...is that the only reliable principle?

@Watson That's not really the kind of answer I'm going for. I'm looking for things like "if $A$ and $B$ are torsion free, and $R$ is a such-and-such ring..." — Eric Auld, Aug 20 '16 at 18:28

Pedro · Answer 1 · 2016-08-30T03:57:19.713

2

It is necessary and sufficient there exists a bilinear map $A\times B\to M$ for some other module $M$ where $a\otimes b$ has nonzero image.

edited Aug 30 '16 at 03:57

answered Aug 19 '16 at 04:53

Pedro

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(And this a necessary and sufficient condition, in fact. Indeed, one can take M to be the tensor product of A and B and the map the canonical bilinear map) – Mariano Suárez-Álvarez Aug 19 '16 at 04:57

When the tensor product of two module elements is nonzero

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