David, I have found this series of video lectures helpful. It is Fredric Schuller's course on Geometrical Anatomy of Theoretical Physics https://www.youtube.com/playlist?list=PLPH7f_7ZlzxTi6kS4vCmv4ZKm9u8g5yic Lecture 11 is Tensor Space Theory II: over a ring. Although you may know all that he says, and I don't think he provides the intuitions you are looking for. Also, I'm intrigued by this book by Daniel A. Fleisch, "A Student's Guide to Vectors and Tensors" http://www.cambridge.org/gb/academic/subjects/physics/mathematical-methods/students-guide-vectors-and-tensors?format=PB&isbn=9780521171908 But I don't think it covers rings or modules.