No. Anyway, do you know the answer? Because the same question was raised in your class and possibly finally they told you the answer.

By the way, I'm new here, I didn't know that here is filled with PhD students and professors that ask really sophisticated questions. I'm just a 2nd-year undergraduate student, for me, tensors, is the highest level of mathematics that I've ever encountered! So, please, go easy on me and explain things to me in the simplest way possible.

Thanks for replying. I'm sorry if the title sounds awkward to you, but the title isn't supposed to mean that all modules have a basis, I do know that some modules can not have a basis, for example $\mathbb{Z_2}$ is not free as a $\mathbb{Z_6}$-module. I'd like to know the structure of these tensor products as R-modules.