Are the following modules $M=\mathbb{Z^3}/\langle(2,1,0),(0,-1,0)\rangle$ and $N=\mathbb{Z^3}/\langle(1,0,0),(0,1,0)\rangle$ isomorphic? Are they free? Which rank do they have?
My attempt: I would say that they are isomorphic since when i reduce the matrix made of (2,1,0), (0,-1,0) in RREF i get (1,0,0), (0,1,0), so they would span the same space? But what are elements of these two modules? And which can be a isomorphism between them? And what about their rank and fact that they're free? They are free if they have a base, but i am not sure how can I write the base in this case.
I am thankful for any sort of help or reference to literature or anything
I have serious doubts that RREF is the one you got. First of all, 2 is not invertible in the ring of integers.
You can easily get $(2,0,0)$ and $(0,1,0)$, and then $M\simeq\mathbb Z/2\mathbb Z\times\mathbb Z$ (why?). On the other side, $N\simeq\mathbb Z$ (why?).