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

1. 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.
   1. $\mathrm{Rad}(I) = \{r \in R \mid r^n \in I, \: (n \in \boldsymbol{N})\}$.
   2. $\mathrm{Rad}(I)$ is the intersection of all prime ideals of $R$ that contain $I$.
   3. $\mathrm{Rad}(I)$ is the pre-image 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.
2. For a multiplicative subset $S$ of a commutative unital ring $R$, and an ideal $I$ of $R$, prove that $S^{-1}\mathrm{Rad}(I) = \mathrm{Rad}(S^{-1}I)$.
3. What’s an example of a Noetherian integral domain that is not a PID?
4. 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 $S^{-1}R$.
5. 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.
6. 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 $R$
7. For a commutative unital ring $R$, prove that the set of zero divisors of $R$ is a union of prime ideals.