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.

Ring Theory

1. Prove that a finite integral domain is in fact a field.
2. Recall what it means for an element of a ring to be nilpotent. For a commutative unital ring $R$, prove that the set of nilpotent elements forms an ideal.
3. Prove that if $R$ is commutative and both $a$ and $b$ in $R$ are nilpotent, the their sum $a+b$ is nilpotent. Why do we need $R$ to be commutative?
4. What is an example of an integral domain $R$ and ideals $I$ and $J$ such that $IJ \neq I \cap J$?
5. In a commutative unital ring, prove that maximal ideas are prime. Prove that the converse is true if your ring is a PID.
6. In the category of commutative unital rings, give an example of a
   1. Ring that is not an Integral Domain.
   2. Integral Domain that is not a GCD Domain.
   3. Integral Domain that is not a UFD. (Bonus points if your example is a GCD Domain.)
   4. UFD that is not a PID.
   5. PID that is not a Euclidian Domain.
   6. Euclidean Domain that is not a Field.
7. For a commutative unital ring $R$, an ideal $M$ is maximal if and only if for each $r \in R \setminus M$ there is some $s \in R$ such that $1-rs \in M$.
8. Recall the definition of an idempotent element of a ring and of a central element of a ring. Two elements $a$ and $b$ of a ring are orthogonal if $ab=0$. If $R$ is a unital ring with idempotent element $e$,
   1. then the element $1-e$ is also idempotent,
   2. and if $e$ is a central element of $R$, then $eR$ and $(1-e)R$ are ideals such that $R = eR \times (1-e)R$.
   3. More generally, there are ideals $\{J_i\}_{i \in 1,\dotsc,n}$ of $R$ such that $R$ can be written as an internal direct sum of the $J_i$, i.e. $R = J_1 \oplus \dotsb \oplus J_n$, if and only if $R$ contains orthogonal central idempotents $\{e_i\}_{i \in 1,\dotsc,n}$ such that $e_i + \dotsb + e_n = 1$ and $J_i = e_i R$ for $i \in \{1,\dotsc,n\}$.
   This is called the Peirce decomposition of a ring. ;p
9. Recall the definition of a local ring. Prove that a commutative unital ring $R$ is local if and only if for all $a, b \in R$ we have that $a+b=1$ implies that either $a$ or $b$ is a unit.
10. Prove that $R$ is local if every non-unit of $R$ is nilpotent.
11. For a unital ring $R$ of characteristic $p$, let $a$ be a nilpotent element of $R$. Prove that $a+1$ is unipotent (that some power of $a+1$ equals $1$).
12. What’s an example of an integral domain $R$ with non-maximal ideal $I$ such that $\mathrm{char}R = 0$ but $\mathrm{char}R/I \neq 0$?
13. For a commutative unital ring $R$, suppose that $f = a_n x^n + a_{n-1}x^{n-1} + \dotsb + a_0$ is a zero divisor in $R[x]$. Prove that there exists some $b \in R$ such that $ba_n = ba_{n-1} = \dotsb = ba_0 = 0$.
14. For a commutative unital ring $R$ and polynomial $f = a_n x^n + a_{n-1}x^{n-1} + \dotsb + a_0 \in R[x]$, $f$ is a unit in $R[x]$ if and only if $a_0$ is a unit in $R$ and $a_1, \dotsc, a_n$ are nilpotent.
15. For indeterminates $x$ and $y$ and a field $\boldsymbol{k}$, prove that $(x,y)$ is not a principal ideal of $\boldsymbol{k}[x,y]$.