Fundamental Exercises in Algebra
In studying for the Algebra Qualifying Exam,
these are some exercises you should really really know.
Most are either common questions on past qualifying exams
or popular homework problems chosen from
Hungerford’s Algebra.
Since these exercises are so fundamental,
solutions to many of them can be found
either in John Dusel’s notes,
or in Kayla Murray’s notes,
or somewhere online like
Math Stack Exchange (MathSE).
If you find a solution online,
you should send me a link so I can post it here.
Otherwise, if you think it’ll help you study,
you can type up a solution and send me a PDF to post here.
Or better, you can type up your solution on MathSE
so that other algebra students can easily find it,
add to it, comment on it, etc.
The MathSE community is going through
a bit of a phase right now, though,
so it would be a good idea to read over this
brief guide to posting on MathSE
before writing up your solution there.
Commutative Algebra

Prove that these three characterizations of $\mathrm{Rad}(I)$,
the radical of an ideal $I$ of a commutative unital ring $R$,
are equivalent. The first one is the usual definition.

$\mathrm{Rad}(I) = \{r \in R \mid r^n \in I, \: (n \in \boldsymbol{N})\}$.

$\mathrm{Rad}(I)$ is the intersection
of all prime ideals of $R$ that contain $I$.

$\mathrm{Rad}(I)$ is the preimage of the ideal of nilpotent elements in $R/I$.
It would be a good idea to
prove that $\mathrm{Rad}(I)$ is honestly an ideal of $R$
directly from the first of these characterizations.

For a multiplicative subset $S$ of a commutative unital ring $R$, and an ideal $I$ of $R$,
prove that $S^{1}\mathrm{Rad}(I) = \mathrm{Rad}(S^{1}I)$.

What’s an example of a Noetherian integral domain
that is not a PID?

For a commutative unital ring $R$, let $I$ be a primary ideal of $R$,
which means that for $a,b \in R$ such that $ab \in I$,
either $a \in I$ or $b^n \in I$ for some $n \in \boldsymbol{N}$.
Let $S$ be a multiplicative subset of $R$ such that $S \cap I = \emptyset$.
Prove that $S^{1}I$ is a primary ideal of $S^{1}R$.

For a commutative unital ring $R$ and proper ideal $I$ of $R$,
prove that $I$ is a primary ideal if and only if
the zero divisors in $R/I$ are all nilpotent.

For a commutative unital ring $R$, let $S$ be a saturated
multiplicative subset $R$, so for $x,y \in R$ we have that
if $xy \in S$ then $x,y \in S$.
Prove that $R \setminus S$ is a union of prime ideals of $R$

For a commutative unital ring $R$,
prove that the set of zero divisors of $R$
is a union of prime ideals.