Exercise 109.1.1. Let $A$ be a ring, and ${\mathfrak m}$ a maximal ideal. In $A[X]$ let $\tilde{\mathfrak m}_1 = ({\mathfrak m}, X)$ and $\tilde{\mathfrak m}_2 = ({\mathfrak m}, X-1)$. Show that

## 109.1 Algebra

This first section just contains some assorted questions.

Exercise 109.1.2. Find an example of a non Noetherian ring $R$ such that every finitely generated ideal of $R$ is finitely presented as an $R$-module. (A ring is said to be *coherent* if the last property holds.)

Exercise 109.1.3. Suppose that $(A, {\mathfrak m}, k)$ is a Noetherian local ring. For any finite $A$-module $M$ define $r(M)$ to be the minimum number of generators of $M$ as an $A$-module. This number equals $\dim _ k M/{\mathfrak m} M = \dim _ k M \otimes _ A k$ by NAK.

Show that $r(M \otimes _ A N) = r(M)r(N)$.

Let $I\subset A $ be an ideal with $r(I) > 1$. Show that $r(I^2) < r(I)^2$.

Conclude that if every ideal in $A$ is a flat module, then $A$ is a PID (or a field).

Exercise 109.1.4. Let $k$ be a field. Show that the following pairs of $k$-algebras are not isomorphic:

$k[x_1, \ldots , x_ n]$ and $k[x_1, \ldots , x_{n + 1}]$ for any $n\geq 1$.

$k[a, b, c, d, e, f]/(ab + cd + ef)$ and $k[x_1, \ldots , x_ n]$ for $n = 5$.

$k[a, b, c, d, e, f]/(ab + cd + ef)$ and $k[x_1, \ldots , x_ n]$ for $n = 6$.

Remark 109.1.5. Of course the idea of this exercise is to find a simple argument in each case rather than applying a “big” theorem. Nonetheless it is good to be guided by general principles.

Exercise 109.1.6. Algebra. (Silly and should be easy.)

Give an example of a ring $A$ and a nonsplit short exact sequence of $A$-modules

\[ 0 \to M_1 \to M_2 \to M_3 \to 0. \]Give an example of a nonsplit sequence of $A$-modules as above and a faithfully flat $A \to B$ such that

\[ 0 \to M_1\otimes _ A B \to M_2\otimes _ A B \to M_3\otimes _ A B \to 0. \]is split as a sequence of $B$-modules.

Exercise 109.1.7. Suppose that $k$ is a field having a primitive $n$th root of unity $\zeta $. This means that $\zeta ^ n = 1$, but $\zeta ^ m\not= 1$ for $0 < m < n$.

Show that the characteristic of $k$ is prime to $n$.

Suppose that $a \in k$ is an element of $k$ which is not an $d$th power in $k$ for any divisor $d$ of $n$ for $n \geq d > 1$. Show that $k[x]/(x^ n-a)$ is a field. (Hint: Consider a splitting field for $x^ n-a$ and use Galois theory.)

Exercise 109.1.8. Let $\nu : k[x]\setminus \{ 0\} \to {\mathbf Z}$ be a map with the following properties: $\nu (fg) = \nu (f) + \nu (g)$ whenever $f$, $g$ not zero, and $\nu (f + g) \geq min(\nu (f), \nu (g))$ whenever $f$, $g$, $f + g$ are not zero, and $\nu (c) = 0$ for all $c\in k^*$.

Show that if $f$, $g$, and $f + g$ are nonzero and $\nu (f) \not= \nu (g)$ then we have equality $\nu (f + g) = min(\nu (f), \nu (g))$.

Show that if $f = \sum a_ i x^ i$, $f\not= 0$, then $\nu (f) \geq min(\{ i\nu (x)\} _{a_ i\not= 0})$. When does equality hold?

Show that if $\nu $ attains a negative value then $\nu (f) = -n \deg (f)$ for some $n\in {\mathbf N}$.

Suppose $\nu (x) \geq 0$. Show that $\{ f \mid f = 0, \ or\ \nu (f) > 0\} $ is a prime ideal of $k[x]$.

Describe all possible $\nu $.

Let $A$ be a ring. An *idempotent* is an element $e \in A$ such that $e^2 = e$. The elements $1$ and $0$ are always idempotent. A *nontrivial idempotent* is an idempotent which is not equal to zero. Two idempotents $e, e' \in A$ are called *orthogonal* if $ee' = 0$.

Exercise 109.1.9. Let $A$ be a ring. Show that $A$ is a product of two nonzero rings if and only if $A$ has a nontrivial idempotent.

Exercise 109.1.10. Let $A$ be a ring and let $I \subset A$ be a locally nilpotent ideal. Show that the map $A \to A/I$ induces a bijection on idempotents. (Hint: It may be easier to prove this when $I$ is nilpotent. Do this first. Then use “absolute Noetherian reduction” to reduce to the nilpotent case.)

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