I am delving into the concept of tensor products and their universal properties in the context of modules over rings. While the universal property of tensor products is well-established for vector spaces over fields, I am seeking to deepen my understanding of how this property translates when dealing with modules over any arbitrary ring, not necessarily a field.
Specifically, I am interested in the following aspects: How is the universal property of tensor products defined for modules over a ring that may include zero divisors or not be a commutative ring?
I would greatly appreciate detailed explanations or references to literature that could help illuminate these points. Any examples that elucidate the nuances of the universal property in this broader context would also be of immense value.
Since you're partly asking for a reference, a very user friendly introduction to tensor products of modules is Conrad, Tensor products. It contains lots of practical tips on how to use the universal property, along with common mistakes, and elaborates on some of the things that hold for tensor products of vector spaces, but not for tensor products of modules.