Let M be an A-module, and a be an ideal of A. Then we denote by M/aM the A/a-module obtained from the quotient of M by the submodule generated by ax for a∈a and x∈M.
This module can be thus defined through the extension of scalars to A/a from the A-module M
So I just learnt what extension of scalars and tensor products of modules are, after seeing this in a textbook, and I was wondering if somebody could give me a few concrete examples of the above situation, so I can grasp what's going on. For example did they have to quotient by aM before extending scalars? I wouldn't have thought so, since every A-Module should be an A/a-module by extension of scalars using the natural quotient map.
So I imagine this is a special case? When it has a particularly straightforward structure? I'm very unfamiliar so I was hoping somebody could give me some pointers
EDIT: I see they were saying that these two are identical. But nevertheless, since I'm unfamiliar with scalar extension, if anybody has examples of extending scalars to modules over the quotient ring I would find it very helpful.