How to show :/⊗r→/ is a well-defined R/I module homomorphism

Question

I'm new to tensor products, and am struggling with showing that f is a well-defined R/I homomorphism, where I is an ideal of R ring, and A is a left R module. Showing scalar multiplication isn't too bad , as f((x+I)(r+I X a)) = f(xr + I X a) = xra + IN = (x+I) (ra+IN) = (x+I) f(ra+ I X a), but am unsure how to show addition.

Answer 1

See Theorem 4.5 here, where $R$ is a commutative ring. Every time you want to build a linear map out of a tensor product, you should use the universal property of tensor products. It lets you avoid trying to develop some special argument to deal with addition, which you're struggling over.

I advise learning how to work with tensor products of modules over a commutative ring before allowing the ring $R$ to be noncommutative and having to bother with left vs. right issues. This topic already has many subtleties even when $R$ is commutative.