Homework 8

Problem 1

Suppose $R$ is a commutative ring. Prove that for each cyclic $R$ -module $M$ we have $T (M) ≃ S (M)$ ; i.e., the tensor algebra is already commutative.

Hint

Show the ideal $C (M)$ is trivial.

Problem 2

Suppose $R$ is a commutative ring and $M$ is a free $R$ -module of rank $n$ , i.e., $M ≃ F (X)$ for some set $X$ with $n$ elements. Prove that $\overset{k}{⋀} (M)$ is a free $R$ -module of rank $(\binom{n}{k})$ for $k = 0, 1, 2, \dots, n$ .

Hint

You can use the fact that for $0 \leq k \leq n$ the $R$ -module $\overset{k}{⋀} (M)$ is a free $R$ -module with a basis explicitly given in terms of a basis for $M$ .

Problem 3

Suppose $F$ is a field of characteristic not 2. Show that for every $F$ -vector space $V$ we have an $F$ -vector space isomorphism
$V \otimes_{F} V ≃ S^{2} (V) \oplus \overset{2}{⋀} (V) .$
In other words, show that every 2-tensor may be written uniquely as a sum of a symmetric and an alternating tensor.

Hint

For each 2-tensor $z$ , show that $z = \frac{1}{2} Sym (z) + \frac{1}{2} Alt (z)$ , where $Sym (z)$ and $Alt (z)$ are the symmetrization and skew-symmetrizationof $z$ , respectively.

This will show $V \otimes_{F} F$ is the sum of the submodules of symmetric and alternating tensors. To show it is the direct sum, verify that the only 2-tensor that is both symmetric and alternating is $0$ .

Finally, you can use the fact that there is an isomorphism between the symmetric power (respectively, exterior power) and the submodule of symmetric tensors (respectively, alternating tensors).

Problem 4

Let $R$ be an integral domain and $M$ be a $R$ -module.

Suppose that $M$ has rank $n$ and $S = {m_{1}, \dots, m_{n}}$ is a maximal linearly independent set of elements in $M$ . Let $N$ be the submodule generated by $S$ . Prove that $N ≃ R^{n}$ and $M / N$ is a torsion $R$ -module.
Conversely, prove that if $M$ contains a submodule $N$ that is free of rank $n$ such that the quotient $M / N$ is a torsion $R$ -module, then $M$ has rank $n$ .

Hints

Show that for any $m \in M$ there is a nonzero $r \in R$ such that $r m = r_{1} m_{1} + \dots + r_{n} m_{n}$ for some $r_{i} \in R$ .
Let ${m_{1}, \dots, m_{n + 1}}$ be a set of $n + 1$ elements of $M$ . Find some nonzero $r_{i} \in R$ so that $r_{i} m_{i}$ can be written using a basis for $N$ . Then show the $r_{i} m_{i}$ (and hence $m_{i}$ ) are linearly dependent.

Problem 5

Let $R$ be an integral domain. Prove that if $A$ and $B$ are $R$ -modules of ranks $m$ and $n$ , respectively, then $A \oplus B$ is an $R$ -module of rank $m + n$ .

Hint

Use an alternate characterization of rank.

Problem 6

Let $R$ be an integral domain, $M$ an $R$ -module, and $N$ a submodule of $M$ . Prove that the rank of $M$ is the sum of the ranks of $N$ and $M / N$ .

(You may assume $M$ has finite rank.)

Hints

For part (2), let ${m_{1} + N, \dots, m_{s} + N}$ be a maximal $R$ -linearly independent set in $M / N$ and ${n_{1}, \dots, n_{l}}$ be a maximal $R$ -linearly independent set in $N$ . Show that the set ${m_{1}, \dots, m_{s}, n_{1}, \dots, n_{l}}$ is an $R$ -linearly independent set in $M$ . Then use an alternate characterization of rank.

Problem 7

Suppose $R$ is an integral domain and $M$ is a $R$ -module.

Show that if $m$ is a nonzero torsion element in $M$ , then the set ${m}$ is $R$ -linearly dependent. Conclude that the rank of $Tor (M)$ is 0.
Show that the rank of $M$ is the same as the rank of the quotient $M / Tor (M)$ .

Problem 8

Let $R$ be an integral domain and $I \subseteq R$ be a non-principal ideal. Prove that $I$ is torsion free of rank 1, but $I$ is not a free $R$ -module.

Hint

To show that the rank of $I$ is no more than 1, observe that for any pair of nonzero elements $a, b \in I$ one has $b \cdot a + (- a) \cdot b = 0$ and so the set ${a, b}$ is $R$ -linearly dependent.