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.

Modules

1. For unital ring $R$, recall what it means for a unitary $R$-module to be simple. Prove that a simple $R$-module $M$ must be cyclic, and that the ring $\mathrm{End}_R(M)$ is a division ring. What about the converse? Is it true that if $\mathrm{End}_R(M)$ is a division ring then $M$ must be simple? (Solution)
2. For a commutative unital ring $R$ and left $R$-modules $M$ and $N$, does $\mathrm{Hom}_R(M,N)$ have any sort of $R$-module structure? Is it necessary to assume that $R$ is commutative? What if $M$ is a right $R$-module instead? (MathSE)
3. For a ring $R$, consider the commutative diagram $$ \require{AMScd} \begin{CD} 0 @>>> A @>{f_1}>> B @>{f_2}>> C @>>> 0\\ @. @. @V{\phi_2}VV @. @.\\ 0 @>>> X @>{g_1}>> Y @>{g_2}>> Z @>>> 0\\ \end{CD} $$ in the category of $R$-modules such that the top and bottom rows are exact. Suppose that there is a some map $\phi_1 \in \mathrm{Hom}_R(A,X)$ such that $\phi_2 f_1 = g_1 \phi_1$. Prove that there exists some map $\phi_3 \in \mathrm{Hom}_R(C,Z)$ such that $\phi_3 f_2 = g_2 \phi_2$.
4. Suppose that $P$ is a projective $R$-module, and is the homomorphic image of some $R$-module $M$. Prove that $P$ is isomorphic to a direct summand of $M$. What is the analogous fact to this one concerning injective $R$-modules?
5. For a unital ring $R$, in the category $R\text{-Mod}$, a free module is projective.
6. More generally than the previous problem, consider the three following adjectives that could describe an $R$-module: $$ \text{free}\qquad\text{projective}\qquad\text{torsion-free} $$ Which of these properties of an $R$-module imply another, and which don’t? Provide proofs and counterexamples.
7. Prove that a direct sum of $R$-modules $\bigoplus_{i \in I} P_i$ is projective if and only if each $P_i$ is projective.
8. Prove that $\boldsymbol{Q}$ is not a projective $\boldsymbol{Z}$-module. What is an example of a projective $\boldsymbol{Z}$-module?
9. Recall the definition of a $\boldsymbol{Z}$-module (abelian group) being divisible. Prove that a unitary $\boldsymbol{Z}$-module is injective if an only if it is divisible.
10. Suppose that in the category $R$-mod, for any object $D$ the functor $\mathrm{Hom}_R(D,-)$ preserves the exactness of the sequence $$0 \longrightarrow A \longrightarrow B % \longrightarrow C \longrightarrow 0\,.$$ Prove that this sequence must split. Prove the converse of this statement too.
11. For a unital ring $R$ and a unitary left $R$-module $M$, write out the details of the left $R$-module isomorphism $M \simeq \mathrm{Hom}_R(R,M)$.
12. For a left $R$-module $M$, write down the details of the natural homomorphism of $R$-modules $\theta_M \colon A \to A^{\ast\ast}$. Prove that $\theta_M$ is an isomorphism if $R$ is unital and $M$ is free with finite basis over $R$.
13. For a homomorphism of left $R$-modules $f \colon M \to N$, write down the details of the natural map $f^\ast \colon M^{\ast\ast} \to N^{\ast\ast}$ such that the following diagram commutes: $$ \require{AMScd} \begin{CD} M @>{\theta_M}>> M^{\ast\ast}\\ @V{f}VV @VV{f^\ast}V\\ N @>{\theta_N}>> N^{\ast\ast}\\ \end{CD} $$
14. For a unital ring $R$ and a unitary left $R$-module $M$, write out the details of the left $R$-module isomorphism $R \otimes_R M \simeq M$.
15. For integers $m$ and $n$, write out the details of the $\boldsymbol{Z}$-bimodule isomorphism $\boldsymbol{Z}/(m) \otimes_\boldsymbol{Z} \boldsymbol{Z}/(n)% \simeq \boldsymbol{Z}/(m,n)\,.$ (MathSE)
16. Let $S$ be a two-sided ideal of a ring $R$ and let $SM$ denote the abelian subgroup of an $R$-module $M$ generated by elements of the form $sm$ for $s \in S$ and $m \in M$. Show that $SM$ is an honest submodule of $M$, describe the natural left $R$-module structure on $(R/S) \otimes_R M$, and show that $(R/S) \otimes_R M \simeq M /SM$ as left $R$-modules.
17. Suppose that $A$ and $A'$ are left $R$-modules and $B$ and $B'$ are right $R$-modules. Take $f \in \mathrm{Hom}(A,A')$ and $g \in \mathrm{Hom}(B,B')$. Is it necessarily true that $$\mathrm{Ker}(f \otimes g) \simeq (\mathrm{Ker}f \otimes B) + (A \otimes \mathrm{Ker}g)\,\text{?}$$
18. Give examples of a commutative ring $R$, of $R$-modules $M$, $M'$, and $N$, and of a map $f \in \mathrm{Hom}(M,M')$ such that
    1. $f$ is injective, but $1 \otimes f \colon N \otimes M \to N \otimes M'$ is not injective.
    2. $f$ is surjective, but $f_\ast \colon \mathrm{Hom}(N,M) \to \mathrm{Hom}(N,M')$, where $f_\ast(h) = f \circ h$, is not surjective.
19. For a ring $R$ and left $R$-modules $M$ and $N$, write down the details of the homomorphism of abelian groups $$M^\ast \otimes_R N \longrightarrow \mathrm{Hom}_R(M,N)\,.$$ Prove that this homomorphism is in fact an isomorphism if $R$ is a field and $M$ and $N$ are finite-dimensional vector spaces over $R$.
20. Let $R$ be an integral domain. For an $R$-module $M$, define $\tau(M) = \{m \in M \mid \mathcal{O}_m \neq \emptyset\}$, where $\mathcal{O}_m$ is the annihilator of $m$ in $R$. Prove that $\tau$ induces a left-exact functor from $R$-mod to the category of torsion $R$-modules, where $M \mapsto \tau(M)$ and $f \mapsto f|_{\tau(M)}$. Why do we need the assumption that $R$ is an integral domain?
21. Let $R$ be a PID, and let $M$ be a unitary left $R$-module. For $s \in R$ recall the definition of a couple of our favorite submodules of $M$: $$ sM = \{sm \mid m \in M\} \qquad M[s] = \{m \in M \mid sm=0\}$$ Let $p$ be a prime element of $R$. Additionally, recall the definition of a cyclic $R$-module, and let $N$ be a cyclic $R$-module of order $r \in R$.
    1. What is the natural way to define $M/pM$ as a vector space over $R/(p)$?
    2. What is the natural way to define $M[p]$ as a vector space over $R/(p)$?
    3. Supposing $s$ is relatively prime to $r$, prove that $sN=N$ and $N[s]=0$.
    4. Suppose $s$ divides $r$, so there is some $k$ such that $sk=r$. Prove that $sN \simeq R/(k)$ and $N[s] \simeq R/(s)$.