# Categorical noetherian-ness

A ring $A$ is noetherian iff the category $_A\mathrm{mod}$ of finitely generated $A$-modules is a full abelian subcategory of the category $_A\mathrm{Mod}$ of all $A$-modules.

# Be rational

Most of representation theory results work out for algebraically closed fields. For instance, we can classify irreducible representations for the dihedral group $D_{2n}$ with $2n$ elements over $\mathbb{C}$. Classify all irreducible representations for $D_{2n}$ over $\mathbb{Q}$.

# Fundamental theorem of algebra…?

Let $C_{n}$ be the cyclic group of order $n$ and consider $\mathbb{C}C_n$ the group algebra. Prove that the number of solutions to the equation $x^k = x$ for $x\in\mathbb{C}C_n$ is $k^n$ for $k>1$.

# Infinite dimensional Schur

Let $A$ be an algebra over $\mathbb{C}$ and let $V$ be an irreducible representation of $A$ with at most countable basis. Then any homomorphism of representations $\phi:V\to V$ is a scalar operator.

# Small rings

Any field that is finitely generated as a ring is a finite field.

# Spooky algebra II

Exhibit a group $G$, a set of generators $X$ for $G$ and a subgroup $H$ such that $xHx^{-1} \subseteq H$ for all $x\in X$ and yet $H$ is not normal in $G$.

# A two-liner

Is there a ring $A$ such that there are two different ring injections $\Bbb{R}\to A$?