If $F$ is a free module then $-\otimes F$ is an exact functor
40996
Proving that free modules are flat (without appealing projective modules)
123242
Half exact functor which is neither right exact nor left exact
28518
Hom of finitely generated modules over a noetherian ring
133238
Why does $\mathrm{Tor}_0^R(M,N)\cong M\otimes_R N$?
101473
Tensor product of a sheaf of module and a residue field
74445
Noetherian and Artinian modules
153852
Example of modules that are projective but not free; torsion-free but not free
37570
$M + N$ and $M \cap N$ are finitely generated $A$-modules implies $M$ and $N$ finitely generated using exact sequences
34065
Submodule of free module over a p.i.d. is free even when the module is not finitely generated?
10679
Equivalence of Categories and of Their Functor Categories
43456
Can the category of chain complexes be realized as a functor category?
7870
For $M\otimes_R N$, why is it so imperative that $M$ be a right $R$-module and $N$ a left $R$-module?
124864
A not free $\mathbb{Z}$-module
87750
Hom / tensor adjunction for $O_X$ modules?
31729
Tensor products of simple modules over algebras
99198
Principal ideal and free module
8374
How to prove that $R/I \otimes_R M \cong M / IM$