Let be a ring epimorphism. Suppose is an -linear mapping between -modules, where the -linear structure is given by . Does it follow that is -linear?
Let be a commutative ring, ideals. If as -modules, then .
Let be the symmetric random walk in –, starting at the origin. So, and the transition probabilities are if and otherwise.
Let be the orthogonal projections of the random walk onto the diagonals and .
- Prove that are independent symmetric random walks in such that if and only if .
- Prove that , as , where and . Conclude that . This is equivalent to the recurrence of the random walk.
Let be an integral domain of finite type over , whose quotient field is of characteristic zero. Let be an element such that at every closed point the image of in the residue field lies in the prime field (where . Prove that must be in .
Prove that the planar Brownian motion is neighbourhood recurrent, but not point recurrent. Conclude that the path of planar Brownian motion is dense in the plane.