Hi. I'm trying to prove this "little fact": let M, N be finitely generated modules over a PID. Then if M+M=N+N (where = means isomorphism and + means direct sum) then M=N.
I'm sure it can be done with the structure theorem (it is obvious from the hypotheses); it looks like it should be...
Well, I managed to do it differently. I didn't realize the exercise preceding this one was going to help me :P
This other exercise said that if S is a connected surface, f: S->R a differentiable function, and the differential of f is always 0, then f is a constant function. This is easily...
Homework Statement
Show that if all normals to a connected surface pass through the origin, the surface is contained in a sphere.
Homework Equations
The Attempt at a Solution
I know a surface is locally the graph of a differentiable function, so in a neighbourhood of a point p, the...