# Dense rings

Let $f:A\rightarrow B$ be a ring epimorphism. Suppose $\varphi:M_1\to M_2$ is an $A$-linear mapping between $B$-modules, where the $A$-linear structure is given by $f$. Does it follow that $\varphi$ is $B$-linear?

# Spooky algebra

Let $A$ be a commutative ring, $I, J\subseteq A$ ideals. If $A/I \simeq A/J$ as $A$-modules, then $I=J$.

# The plane is too small 2

Let $\{X_n|n\in\mathbb{N}_0\}\subset\mathbb{Z}^2$ be the symmetric random walk in $2$$d$, starting at the origin. So, $X_0=(0,0)$ and the transition probabilities are $p_{i,j}=1/4$ if $|i-j|=1$ and $0$ otherwise.

Let $X_n^{+},X_n^{-}$ be the orthogonal projections of the random walk onto the diagonals $\{y=x\}$ and $\{y=-x\}$.

1. Prove that $X_n^{+},X_n^{-}$ are independent symmetric random walks in $\mathbb{Z}/2$ such that $X_n=0$ if and only if $X_n^{+}=0=X_n^{-}$.
2. Prove that $p_{00}^{(2n)} = (\binom{2n}{n}2^{-2n})^2 \sim \frac{A}{n}$, as $n\to\infty$, where $A \in \mathbb{R}_{>0}$ and $p_{00}^{(2n)}=P(X_{2n}=(0,0)|X_0=(0,0))$. Conclude that $\sum_{n=0}^{\infty} p_{00}^{(n)} = \infty$. This is equivalent to the recurrence of the random walk.

# Local to Global

Let $R$ be an integral domain of finite type over $\mathbb{Z}$, whose quotient field is of characteristic zero. Let $\alpha\in R$ be an element such that at every closed point $\mathfrak{p}$ the image of $\alpha$ in the residue field $R/\mathfrak{p}$ lies in the prime field $\mathbb{F}_p$ (where $p = \mathfrak{p}\cap\mathbb{Z})$. Prove that $\alpha$ must be in $\mathbb{Q}$.

# The plane is too small

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.