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A First Course in Noncommutative Rings

Tsit-Yuen Lam

Chapter 7

Local Rings, Semilocal Rings, and Idempotents - all with Video Answers

Educators

Section 19

Local rings

Problem 1

Show that the opposite ring of a local ring is also a local ring.

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Problem 2

For any field $k$, show that the $3 \times 3$ matrices in (19.15) form a commutative local ring $R$ whose maximal ideal has square zero. Check that $R \cong \operatorname{End}_{k G} V$ in the example (19.15).

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Problem 3

What can you say about a local ring which is von Neumann regular?

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Problem 4

(This exercise refines (19.10).) Let $R=k G$ where $k$ is a field and $G$ is a nontrivial finite group. Show that the following statements are equivalent: (1) $R$ is a local ring. (2) $R_R$ is an indecomposable $R$ module. (3) $R / r$ ad $R$ is a simple ring. (4) $k$ has characteristic $p>0$ and $G$ is a $p$-group.

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Problem 5

Let $\mathfrak{A}$ be an ideal in a ring $R$ such that $\mathfrak{A}$ is maximal as a left ideal. Show that $R / \mathfrak{A}^n$ is a local ring for every integer $n \geq 1$.

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Problem 6

Show that if a ring $R$ has a unique maximal ideal, then the center $Z(R)$ of $R$ is a local ring. (In particular, the center of a local ring is a local ring.)

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Problem 7

A domain $R$ is called a right discrete valuation ring if there is a nonunit $\pi \in R$ such that every nonzero element $a \in R$ can be written in the form $\pi^n u$ where $n \geq 0$ and $u$ is a unit. Show that (1) $R$ is a local domain, (2) every right ideal in $R$ has the form $\pi^i R$ for some $i \geq 0$, (3) each $\pi^i R$ is an ideal of $R$, and (4) $\bigcap_{i \geq 1} \pi^i R=0$. Give an example of a noncommutative right discrete valuation ring by using the twisted power series construction.

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Problem 8

(Brungs) Let $R$ be a nonzero ring such that any collection of right ideals in $R$ has a largest member (i.e., one that contains all the others). Show that (1) $R$ is a local ring, (2) every right ideal of $R$ is principal, and is an ideal. (Hint. For (2), let $I$ be a nonzero right ideal, and let $I^{\prime}$ be the largest right ideal properly contained in $I$. Show that $I=a R$ for any $a \in I \backslash I^{\prime}$. If there exists a right ideal which is not an ideal, consider the largest one and get a contradiction.)

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Problem 9

For a division ring $D$ and a (not necessarily abelian) ordered group $(G,<)$, a function $v: D^* \rightarrow G$ is called a valuation if $v(a b)=v(a) v(b)$ for all $a, b \in D^*$ and $v(a+b) \geq \min \{v(a), v(b)\}$ for all $a, b \in D^*$ such that $a+b \neq 0$. Given such a valuation, let

$$
R=\{0\} \cup\left\{a \in D^*: v(a) \geq 1\right\} .
$$

(1) Show that $R$ is a local ring. (2) Show that any left or right ideal in $R$ is an ideal. (3) Show that any finitely generated left ideal in $R$ is principal. (4) Show that any collection of right ideals in $R$ has a largest member. (5) Show that $a R a^{-1}=R$ for any $a \in D^*$. (6) Show that for any $a \in D^*$, either $a \in R$ or $a^{-1} \in R$.

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Problem 10

Let $R$ be a subring of a division ring $D$ which satisfies the two properties (5), (6) in Exercise 9. Show that there exists an ordered group $(G,<)$ and a valuation $v: D^* \rightarrow G$ such that

$$
R=\{0\} \cup\left\{a \in D^*: v(a) \geq 1\right\} .
$$

(Such a subring $R$ of a division ring $D$ is called an invariant valuation ring of $D$. If $D$ is a field, the property (5) is automatic; in this case, we get back the usual (commutative) valuation rings.)

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Problem 11

Deduce the fact that a finitely generated projective right module over a local ring is free (Theorem (19.29)) from the Krull-SchmidtAzumaya Theorem.

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Problem 12

Let $k$ be a field with the property that $k(t)$ (the rational function field in one variable over $k$ ) is isomorphic to $k$. (For instance, $\mathbb{Q}\left(x_1, x_2, \ldots\right)$ is such a field $k$.) Let $\theta: k(t) \rightarrow k$ be a fixed isomorphism. Let $p(t)$ be a fixed irreducible polynomial in $k[t]$, and let $A$ be the discrete valuation ring obtained by localizing $k[t]$ at the prime ideal $(p(t))$. On $R=A \oplus A$, define a multiplication by $(a, b)\left(a^{\prime}, b^{\prime}\right)=\left(a a^{\prime}, b a^{\prime}+\theta(a) b^{\prime}\right)$. Then $R$ is a ring with identity $(1,0)$. (1) Show that $R$ is a local ring with $\mathrm{rad} R=A \cdot p(t) \oplus A$. (2) Show that $\bigcap_{i=1}^{\infty}(\mathrm{rad} R)^i=(0) \oplus A$, and that this is the prime radical of $R$. (3) Show that every right ideal of $R$ is an ideal. (4) Show that $R$ is right noetherian but not left noetherian. (This exercise is to be contrasted with Krull's Theorem in commutative algebra which states that, for any commutative noetherian local ring $S, \bigcap_{i=1}^{\infty}(\operatorname{rad} S)^i=(0)$.)

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Problem 13

Show that any finitely generated projective right module over a right artinian ring $R$ is isomorphic to a finite direct sum of principal indecomposable right modules of $R$.

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Problem 14

Give an example of a local ring whose unique maximal ideal is nil but not nilpotent.

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Problem 15

Let ( $R, J$ ) be a local ring, and $M$ be a finitely generated left $R$ module. If $\operatorname{Hom}_R(M, R / J)=0$, show that $M=0$. (Hint. Note that $H o m_R(M / J M, R / J) \rightarrow H o m_R(M, R / J)$ is injective.)

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Problem 16

Let ( $R, J$ ) be a left noetherian local ring, and $M$ be a finitely generated left $R$-module. Show that $M$ is a free $R$-module iff, for any exact sequence $0 \rightarrow A \rightarrow B \rightarrow M \rightarrow 0$ of left $R$-modules, the induced sequence $0 \rightarrow A / J A \rightarrow B / J B \rightarrow M / J M \rightarrow 0$ remains exact.

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