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.