# 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 $\Bbb{R}\to A$?

# 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 $I=J$.

# Splitting Q

Are there any decomposable quotients of $\Bbb{Q}$ (viewed as a $\Bbb{Z}$-module)?

# Tardily projective? You’re free then

Suppose $A$ is a $\mathbb Z$-graded connected algebra over a field $k$ concentrated in positive degrees, and suppose $P$ is a projective graded $A$-module concentrated in degrees above some $m\in \mathbb Z$ . Then $P$ is actually a free $A$-module.

Under the same hypothesis over $A$, show that if $M$ is a graded module and $M\otimes_A k=0$ where $k$ is the quotient of $A$ by the ideal of positively graded elements, then $M=0$.

Using the above, show that if $M$ is graded and $\mathrm{Tor}_A^1(M,k)=0$, then $M$ is free. To illustrate, it follows projective graded modules over polynomial algebras are free.