Suppose that $A,B,C$ are $R$-modules for $R$ a commutative unital ring, and
$$ \mathcal{B} : A \times B \to C, $$
is an $R$-bilinear form inducing a map on the tensor product
$$ \tilde{\mathcal{B}} : A \otimes_R B \to C, $$
where
$$ \tilde{\mathcal{B}}\left( \sum_i \left(a_i \otimes_R b_i\right)\right) = \sum_i \mathcal{B}(a_i,b_i). $$
Then is there a general method to characterise $\ker\tilde{\mathcal{B}}$?
We can make the following simple statement: If
$$ \iota : A \times B \to A \otimes_R B, $$
is the canonical homomorphism, then we know that $\mathcal{B} = \tilde{\mathcal{B}}\circ \iota$, so
$$ \ker \mathcal{B} = \ker{\iota} + \iota^{-1}\left( \ker\left( \tilde{\cal{B}}_{\big \vert \iota(A \times B)} \right) \right), $$
and so in particular
$$ \ker\left( \tilde{\cal{B}}_{\big \vert \iota(A \times B)} \right) = \iota(\ker\mathcal{B}), $$
but can we make any statements more general than this? Any comments or even just reference suggestions would be greatly appreciated.