I have recently learned about tensor products of modules,specifically the material in Dummit and Foote chapter 10 section 4. My understanding is that the construction of tensor spaces is important because they are a natural staging area for studying R-multilinear functions on a product of modules.
In linear algebra, I know of a few multilinear functions that are useful: like the inner product or the determinant, and also forms and metrics on manifolds. However, I've never really seen how the language of tensor products of vector spaces is really helpful in working with those two examples. Further, and more importantly, I can't think of any examples of why I would need to think about a bilinear or multilinear function on an R-module where R isn't a field.
Could anyone provide a tangible result that is simplified by using these constructions and especially some motivation for why I should care about R-multilinear functions when R isn't a field? In particular, how do the concepts of tensor products connect with other areas of mathematics or applications.