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If I want to define a homomorphism, $f$, from $A\otimes_R B$ into some $R$ module $M$. If I defined it on simple tensors $a\otimes b$ what are the conditions I need to check to make this is well defined.

Does it suffice to check that $f(r(a\otimes b))=f((ra)\otimes b)=f(a\otimes (rb))$ or is it more complicated than that.

Thank you all.

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up vote 4 down vote accepted

You simply need to define an $R$-bilinear map $\tilde{f} : A \times B \rightarrow M$. The universal property of the tensor product then induces an $R$-module homomorphism $f : A \otimes_R B \rightarrow M$. To check that $\tilde{f}$ is $R$-bilinear, you must show:

(1) $\tilde{f}(ra,b) = \tilde{f}(a,rb) = r\tilde{f}(a,b)$

(2) $\tilde{f}(a_1 + a_2,b) = \tilde{f}(a_1,b) + \tilde{f}(a_2,b)$

(3) $\tilde{f}(a,b_1 + b_2) = \tilde{f}(a,b_1) + \tilde{f}(a,b_2)$

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