3.1 Modules

Know basic notions:

(i) for a general, not necessarily commutative ring $R$ :
$R$-module, submodule, quotient module, isomorphism theorem(s), correspondence theorem; annihilators and torsion elements; cyclic, simple (or irreducible) and free modules; direct sums and direct products and their universal properties; know that the direct product of infinitely many copies of $\mathbb{Z}$ is not a free $\mathbb{Z}$-module; spanning and linearly independent subsets; sets of $R$-module homomorphisms and the algebraic structures on them: $\operatorname{Hom}_{R}(M, N)$ is always an abelian group and an $R$-module if $R$ is commutative; $\operatorname{End}_{R}(M)=\operatorname{Hom}_{R}(M, M)$ is always a ring and an $R$-algebra if $R$ is commutative; know that $\operatorname{End}_{R}\left(R^{n}\right)$ is isomorphic to $M_{n}(R)$ if $R$ is commutative;
(ii) for an integral domain $R$ : torsion submodule; well-defined rank of a free $R$-module; know that submodules of free modules need not be free but that rk $M^{\prime} \leq \operatorname{rk} M$ if $M^{\prime}$ is a free submodule of a free module $M$.

Know the basic theorems for finitely generated modules over a PID $R$ : submodules of a free $R$-module $M$ are free (this is also true if $M$ is not finitely generated); torsion free finitely generated $R$-modules are free; a finitely generated $R$-module $M$ is always the direct some of its torsion submodule $T(M)$ and a free (finitely generated) submodule, and $T(M)$ is a finite direct sum of cyclic modules; existence and uniqueness of invariant factors and elementary divisors, namely (see also Section 1.3):

Elementary Divisor Form (longest decomposition into cyclics): Any finitely generated $R$-module $M$ can be written as $M=R /\left(p_{1}^{e_{1}}\right) \oplus \ldots R /\left(p_{r}^{e_{r}}\right) \oplus R^{n}$, where $n$ is the rank of $M / T(M)$, the $p_{i}$ are (not necessarily distinct) prime elements of $R$, determined up to multiplication with units by $M$, and the $e_{i}$ are natural numbers, also uniquely determined by $M$ (up to possible permutations of the direct summands). The prime powers $p_{i}^{e_{i}}(1 \leq i \leq r)$ are called the elementary divisors of $M$.

Invariant Factor Form (shortest decomposition into cyclics): $M=R /\left(d_{1}\right) \oplus \ldots R /\left(d_{s}\right) \oplus R^{n}$ with $n$ as before and elements $d_{i} \in R \backslash\left(R^{*} \cup\{0\}\right)$ satisfying $d_{1}\left|d_{2} \ldots\right| d_{s}$. These elements $d_{i}$ are, up to multiplication with units, uniquely determined by $M$ and called the invariant factors of $M$.
(Side remark: The number $n$ is often called the Betti number of $M$.)
Know the "Compatible Basis Theorem": If $N$ is a submodule of a finitely generated free $R$-module $M(R$ a PID $)$, then there exist an $R$-basis $y_{1}, \ldots, y_{n}$ of $M$ and elements $d_{i} \in R \backslash\{0\}$ ( $1 \leq i \leq s$ ), up to multiplication with units uniquely determined by $M$ and $N$ and called the invariant factors of $N$ with respect to $M$, such that $d_{1}\left|d_{2} \ldots\right| d_{s}$ and $d_{1} y_{1}, \ldots, d_{s} y_{s}$ is an $R$-basis of $N$.

Know How to Decompose if $R$ is Euclidean:

(I) Write $M=F / N$ for $F$ free on $n$ generators $\left\{y_{1}, \ldots, y_{n}\right\}$, and choose (not necessarily free) generators $c_{i 1} y_{1}+\ldots c_{i n} y_{n}, i=1, \ldots, m$ for $N$ (in other words: $c_{i 1} y_{1}+\ldots c_{i n} y_{n}=0$ with $i=1, \ldots, m$ is a set of defining relations for $M$ ). Define the relations matrix $C=\left(c_{i j}\right)$.
(II) Now apply elementary row and column operations operations to $C$ until you reach a diagonal form $\operatorname{diag}\left(d_{1}, \ldots, d_{s}\right)$ with $d_{i} \in R \backslash\{0\}$ and $d_{1}\left|d_{2}\right| \ldots \mid d_{s}$, meaning that the resulting matrix has $d_{1}, \ldots, d_{s}$ as its first $s$ entries on the main diagonal and zeros everywhere else (the Euclidean Algorithm guarantees that this is possible).
(III) Then the $d_{i}$ ’s which are no units are the invariant factors of $M$, and $M$ is isomorphic to $\bigoplus_{d_{i} \notin R^{*}} R /\left(d_{i}\right) \oplus R^{n-s}$. Keeping track of the column operations, this procedure also yields compatible bases for $M$ and $N$, implying that $d_{1}, \ldots, d_{s}$ (including the units) are the invariant factors of $N$ with respect to $F$.

Related Problems

1. (Sep $82 \# 2$ ) If $a_{1}, \ldots, a_{n}$ in a PID $R$ have $\operatorname{gcd} d$, show that there exists an invertible $n \times n$ matrix $Q$ of determinant 1 over $R$ with $Q\left[a_{1}, \ldots, a_{n}\right]^{T}=[d, 0,0 \ldots 0]^{T}$. (This is false if $R$ is merely a UFD).

2. (Sep $84 \# 5$ ) If $A, B, C$ are submodules of a general $R$-module $M$, with $A \supseteq C$, show $A \cap(B+ C)=(A \cap B)+C$.

3. (Sep 84 #6) Find the invariant factors of the following $3 \times 3$ matrices over $\mathbb{Z}$, and decide if they are equivalent: $\left(\begin{array}{ccc}10 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 6\end{array}\right),\left(\begin{array}{ccc}4 & 6 & 4 \\ 4 & 20 & 12 \\ 20 & 0 & 20\end{array}\right)$.

4. (Sept $86 \# 5$ ) If $M$ is an artinian module over a general $R$, show that any injective endomorphism is surjective.
   (A module $M$ over a general ring $R$ is called artinian if there is no infinite chain of strictly decreasing submodules of $M$. Standard example (explain why this is an example!): $R=$ finite dimensional $K$-algebra for a field $K$ and $M=$ finitely generated $R$-module.)

5. (Jan 87 #6; Aug 88 #6) If $I$ is an ideal of a commutative ring $R$, show
   (a) $R / I$ is simple as an $R$-module $\Longleftrightarrow I$ is a maximal ideal,
   (b) $I$ prime $\Longrightarrow R / I$ is an indecomposable $R$-module;
   (An $R$-module is called indecomposable if it cannot be written as the direct sum of two nonzero $R$-modules.)
   (c) if $I$ is prime, what kind of ring is $R / I$ ?

6. (May $91 \# 6$ ) Let $M=\mathbb{Z} \oplus \mathbb{Z}$ be the free module of rank 2 over the ring $\mathbb{Z}$ of integers. Let $S$ be the submodule of $M$ spanned by $x=(3,0), y=(0,4), z=(6,2)$. Find a $\mathbb{Z}$-basis for the submodule $S$.

7. (Aug 94 #8) If the annihilator of a left $R$-module $M$ is $A n n_{R}(M)=\{a \in R \mid a M=0\}$, show that for submodules $M_{1}, M_{2}$ of $M$ we have $A n n_{R}\left(M_{1}+M_{2}\right)=A n n_{R}\left(M_{1}\right) \cap A n n_{R}\left(M_{2}\right)$. Show furthermore that we have $A n n_{R}\left(M_{1}\right)+A n n_{R}\left(M_{2}\right) \subseteq A n n_{R}\left(M_{1} \cap M_{2}\right)$, but show that this inclusion could be strict.
   Additional question: Is $A n n_{R}\left(M_{1}\right)+A n n_{R}\left(M_{2}\right)=A n n_{R}\left(M_{1} \cap M_{2}\right)$ true if $R$ is a PID and $M$ is a finitely generated torsion $R$-module (no free summand)?

8. (Aug $96 \# 3$ ) Let $M$ be a finitely generated module over a ring $R$. Show that any generating set for $M$ as $R$-module must contain a finite generating set. Conclude that $M$ has a minimal generating set (no proper subset generates $M$ ), and that every minimal generating set of $M$ is finite.

9. (Aug $97 \# 7$ ) Let $M$ be the free module over $\mathbb{Q}[x]$ with basis $v_{1}, v_{2}, v_{3}, N$ a submodule with basis $w_{1}, w_{2}, w_{3}$, where

$$\begin{aligned} & w_{1}=\left(x^{3}-x^{2}-x+1\right) v_{1}+\left(2 x^{2}+2 x\right) v_{2}+\left(x^{3}+x^{2}\right) v_{3} \\ & w_{2}=\left(2 x^{3}-3 x^{2}-3 x+2\right) v_{1}+\left(5 x^{2}+5 x\right) v_{2}+\left(2 x^{3}+2 x^{2}\right) v_{3} \\ & w_{3}=\left(x^{3}-x\right) v_{1}+\left(x^{2}+x\right) v_{2}+\left(x^{3}+x^{2}\right) v_{3} \end{aligned}$$

(a) There is a theorem which implies that $M$ has another basis $u_{1}, u_{2}, u_{3}$ for which $d_{1} u_{1}, d_{2} u_{2}, d_{3} u_{3}$ are a basis for $N$ for some $d_{1}, d_{2}, d_{3} \in \mathbb{Q}[x]$ such that $d_{1}\left|d_{2}\right| d_{3}$. Write a carefully worded statement of this theorem, giving all appropriate hypotheses and conclusions.
(b) Find the values $d_{1}, d_{2}, d_{3}$ for the example given here.
10. (Jan 98 #4) Let $M$ be a module over a ring. A section $A: B$ of $M$ is a pair of submodules $A, B$ of $M$ with $B \subseteq A$; a trivial section is where $B=A$. A submodule $C$ of $M$ covers a section $A: B$ if $(A \cap C)+B=A$, and avoids $A: B$ if $A \cap C \subseteq B$.
(a) Show that $C$ covers $A: B$ iff $A \subseteq B+C$, and avoids $A: B$ iff $A \cap(B+C)=B$.
(b) Show that every $C$ simultaneously covers and avoids any trivial section; that any $C$ with $C \supseteq A$ covers $A: B ;$ and that any $C$ with $C \subseteq B$ avoids $A: B$.
(c) Give an example of a $\mathbb{Z}$-module $M$ and a submodule $C$ that covers one nontrivial section $A: B$ and avoids another nontrivial section $A^{\prime}: B^{\prime}$ for which the two quotients $A / B$ and $A^{\prime} / B^{\prime}$ are isomorphic.
11. (Jan $05 \# 3$ ) Let $R$ be a PID, $p$ a prime element of $R$ and $M \neq\{0\}$ a finitely generated $R$-module such that there exists a natural number $k$ with $p^{k} M=\{0\}$. We choose $k$ minimal
with this property, i.e. $p^{k-1} M \neq\{0\}$.
(a) Describe the structure of $M$. Show that $p^{k}$ is an elementary divisor of $M$ and that each elementary divisor of $M$ divides $p^{k}$.
(b) Let $m$ be any element of $M$ with the property that $p^{k-1} m \neq 0$. Show that the cyclic module $N:=R m$ has a complement $C$ in $M$, i.e. $M=N \oplus C$.
12. (Jan 08 #2) (a) Let $R$ be a commutative ring with 1 , and $M$ a finite direct sum $M= M_{1} \oplus \cdots \oplus M_{n}$ of simple unital left $R$-modules $M_{i}$. Show that $M$ has both the ascending and descending chain conditions on $R$-submodules.
(b) Where does your argument use the hypotheses that $R$ is unital, $R$ is commutative, or $M$ is unital?
13. (Aug $11 \# 3$ ) (a) Let $V$ be an noetherian module for a ring $R$, so that $V$ satisfies the ascending chain condition on submodules. Let $T: V \rightarrow V$ be a surjective $R$-endomorphism. Prove that $T$ is an isomorphism.
(b) In (a) suppose that $R$ is a field, so $V$ is a vector space. Give another explanation of (a) in terms of the rank and nullity of $T$.
14. (Jan $12 \# 6$ ) Let $R$ be a commutative ring with 1 . Recall that a left R -module $M$ is called Noetherian if it satisfies the ascending chain condition on submodules and Artinian if it satisfies the descending chain condition on submodules. Assume that an $R$-module $M$ is both Artinian and Noetherian. (For example, $R$ might be a field, and $M$ might be a finitedimensional vector space over $R$ ). Let $T: M \rightarrow M$ be an $R$-module homomorphism.
(a) Prove that there exists $k \in \mathbb{N}$ s.t. $\operatorname{Ker}\left(T^{k}\right)=\operatorname{Ker}\left(T^{2 k}\right)$ and $\operatorname{Im}\left(T^{k}\right)=\operatorname{Im}\left(T^{2 k}\right)$.
(b) Prove that if $k$ is as in part (a), then $M=\operatorname{Ker}\left(T^{k}\right) \oplus \operatorname{Im}\left(T^{k}\right)$.
(c) Deduce from (a) and (b) that there exist submodules $M_{0}$ and $M_{1}$ of $M$ s.t. $M=M_{0} \oplus M_{1}$, $T_{\mid M_{0}}$ is nilpotent and $T_{\mid M_{1}}$ is invertible (as a map from $M_{1}$ to $M_{1}$ ).
(d) Now assume that $R$ is a field of characteristic zero, $M$ is a finite-dimensional vector space over $R$ and $\operatorname{tr}\left(T^{n}\right)=0$ for every $n \in \mathbb{Z}_{>0}$. Prove that $T$ is nilpotent. Hint: Apply (c), assume that $M_{1} \neq 0$ and reach a contradiction by applying the Cayley-Hamilton theorem to $T_{\mid M_{1}}$.
15. (Aug $15 \# 5$ ) Let $R$ be a commutative ring with identity, and let $I$ be a nilpotent ideal, i.e., $I^{k}=0$ for some $k$. Let $M, N$ be two $R$-modules, and let $f: M \rightarrow N$ be an $R$-homomorphism. Suppose that the induced homomorphism from $M / I M$ to $N / I N$ is surjective. Prove that $f$ is surjective.
16. (Jan $16 \# 3$ ) Let R be a commutative ring with identity. A non-zero $R$-module $M$ is said to be irreducible if 0 and $M$ are the only submodules of $M$. Prove that $M$ is irreducible if and only if $M \cong R / m$, where $m$ is a maximal ideal of $R$.
17. (Aug 16 # 7)
(a)Give a complete and irredundant list of abelian groups of order 144.
(b)Give a complete and irredundant list of finitely generated modules over $\mathbb{F}_{2}[t]$ where the polynomial $t^{4}+t^{3}+t+1$ acts trivially.
18. (Jan 17 #7) Consider the matrix

$$A=\left[\begin{array}{ccc} 12 & 4 & -16 \\ 4 & 3 & -7 \\ 8 & 3 & -11 \end{array}\right]$$

(a) Find the characteristic and minimal polynomials of this polynomial and its Jordan normal form.
(b) Consider map $\mathbb{Z}^{3} \rightarrow \mathbb{Z}^{3}$ induced by $A$. Describe the kernel and cokernel of this map as a sum of copies of $\mathbb{Z}$ and $\mathbb{Z} / n \mathbb{Z}$.
19. (Aug 18 #4) Let $R$ be an integral domain and let $M$ be a nontrivial torsion $R$-module.
(a) If $M$ is finitely generated then the annihilator of $M$ in $R$ is nontrivial. Recall that the annihilator of $M$ is the ideal $\{r \mid r m=0$ for all $m \in M\}$.
(b) Find an integral domain $R$ and a torsion module $M$ over $R$ whose annihilator is the zero ideal.
20. (Jan $19 \# 6$ ) Let $R$ be a commutative ring with 1 , let $M$ be an $R$-module and $N$ a submodule of $R$.
(a) Prove that if $N$ and $M / N$ are both finitely generated, then $M$ is finitely generated
(b) Give an example where $M$ is finitely generated and $N$ is not.
21. (Jan 20 #6) In each part of this problem determine if the given objects are isomorphic:
(a) $\mathbb{R}$ and $\mathbb{C}$ as $\mathbb{Q}$-vector spaces
(b) $\mathbb{C}$ and $\mathbb{R} \times \mathbb{R}$ as rings
(c) $\mathbb{R}[x] /\left(x^{2}-1\right)$ and $\mathbb{R} \times \mathbb{R}$ as rings
(d) $\mathbb{R}[x] /(x-1)$ and $\mathbb{R}[x] /(x+1)$ as $\mathbb{R}[x]$-modules
22. (Jan 22 #6) In each part determine whether the statement is TRUE (in all cases) or FALSE (in at least one case) and prove your claim. An answer (correct or incorrect) without explanation will not receive any credit.
(a) Let $R$ be a commutative domain with 1 and $M$ an $R$-module. If $x \in M$ and $y \in M$ are both torsion elements, then $x+y$ is also a torsion element.
(b) If $K$ and $L$ are fields, then $K \otimes_{\mathbb{Z}} L$ is nonzero.
(c) If $R$ is a commutative ring with 1 and every $R$-module is free, then $R$ is a field.
(d) If $R$ is a commutative ring with 1 and $M$ is a finitely generated $R$-module, then every submodule of $M$ is finitely generated.