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A Course in Ring Theory

Donald S. Passman

Chapter 27

Uniform Dimension - all with Video Answers

Educators

Chapter Questions

03:56

Problem 1

Use Zorn's Lemma to prove that every prime ideal of $R$ contains a minimal prime. Conclude that $R$ is semiprime if and only if the intersection of all its minimal primes is zero.

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
08:25

Problem 2

Let $F$ be a field and let $R$ be the subring of the strong direct sum $\prod_{n=0}^{\infty} F$ consisting of all sequences that are eventually constant. Thus $R$ consists of all elements $\prod_{n=0}^{\infty} a_{n}$ such that $a_{n} \in F$ and $a_{k}=a_{k+1}$ for all sufficiently large $k$. First verify that $R$ is a commutative von Neumann regular ring. Next, let $I$ be the ideal of $R$ consisting of all sequences that are eventually 0 and define the $R$-module homomorphism $\sigma: I \rightarrow R$ by $\prod_{n=0}^{\infty} a_{n} \mapsto \prod_{n=0}^{\infty} f_{n} a_{n}$, where $f_{n}=0$ for $n$ even and $f_{n}=1$ for $n$ odd. Show that $\sigma$ does not extend to an $R-$ homomorphism $\sigma^{*}: R \rightarrow R$ and conclude that $R$ is not self-injective.

Ely Crowder
Ely Crowder
Numerade Educator
02:16

Problem 3

Let $R=R_{1}+R_{2}+\cdots+R_{n}$ be the direct sum of the $n$ rings $R_{i}$. If $D$ is a right ideal of $R$, prove that $D=D_{1}+D_{2}+\cdots+D_{n}$, where
$D_{i}=D \cap R_{i}$ and that $D$ is dense if and only if each $D_{i}$ is dense in $R_{i} .$ Conclude from Theorem $24.8$ that $\mathbf{Q}_{\max }(R)=\cdot \sum_{i} \mathbf{Q}_{\max }\left(R_{i}\right)$. Furthermore, prove that $\mathrm{Q}_{\mathrm{cl}}(R)=\cdot \sum_{i} \mathrm{Q}_{\mathrm{cl}}\left(R_{i}\right)$ if either side of this equality is assumed to exist.

In the next three problems assume that $I \subseteq R \subseteq S$, where $R$ and $S$ are rings and where $I \triangleleft S$ with $\operatorname{lann}_{S}(I)=0$. For example, we could take $S=\mathbb{Z}[\omega]$, where $\omega=(-1+\sqrt{-3}) / 2$ is a primitive cube root of $1, R=\mathbb{Z}[\sqrt{-3}]$, and $I=2 S$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
01:05

Problem 4

Let $E=\mathbf{E}\left(S_{S}\right)$ so that $R \subseteq S \subseteq E .$ Prove that $\operatorname{lann}_{E}(I)=0$ and that $R_{R}$ ess $E_{R}$. Furthermore, show that $E$ is an injective $R$-module. For the latter, first prove that if $J$ is a right ideal of $R$ and if $\sigma: J \rightarrow E$ is an $R$-homomorphism, then \sigmarextends to an $S$-module homomorphism $\sigma^{t}: J S \rightarrow E$

Anthony Ramos
Anthony Ramos
Numerade Educator
01:45

Problem 5

Prove that $\mathbf{Q}_{\max }(R)=\mathbf{Q}_{\max }(S) .$ For this, observe that $E_{R}$ is the injective hull of $R_{R}$ and note that $\operatorname{End}_{R}(E)=\operatorname{End}_{S}(E) .$ Furthermore, if $R$ is prime, deduce that $\mathbf{Q}_{\mathbf{r}}(R)=\mathbf{Q}_{\mathbf{r}}(S)$ and $\mathbf{Q}_{\mathbf{s}}(R)=\mathbf{Q}_{\mathbf{s}}(S)$.

Adriano Chikande
Adriano Chikande
Numerade Educator
03:13

Problem 6

Now assume that $I$ contains a regular element of $S$. Prove that $Q_{c l}(R)=Q_{c l}(S)$ if either of these two quotient rings is assumed to exist.

Let $R$ be a semiprime ring and let $A, B \triangleleft R$. Since $(A \cap B)^{2} \subseteq A B \subseteq$ $A \cap B$, it follows that $A B=0$ if and only if $A \cap B=0$. In particular, $A B=0$ if and only if $B A=0$ and hence $\operatorname{rann}_{R}(A)=\operatorname{lann}_{R}(A)$. We use $\operatorname{ann}_{R}(A)$ to denote this common annihilator. In the next two exercises assume that $P_{1}, P_{2}, \ldots, P_{n}$ are incomparable prime ideals of $R$ with $\bigcap_{1}^{n} P_{i}=0$

Gideon Idumah
Gideon Idumah
Numerade Educator
02:59

Problem 7

Prove that $N_{i}=\operatorname{ann}_{R}\left(P_{i}\right)=\bigcap_{j \neq i} P_{j}$ and that $\operatorname{ann}_{R}\left(N_{i}\right)=P_{i}$. Furthermore, show that $N=\cdot \sum_{i} N_{i}^{\top}$ is an ideal of $R$ with $\operatorname{ann}_{R}(N)=0$

Supratim Pal
Supratim Pal
Numerade Educator
02:57

Problem 8

8. Define $S$ to be the ring direct sum $S=\oplus \sum_{1}^{n} R / P_{i}$ and let $-: R \rightarrow S$ be the natural ring embedding. Prove that $\bar{N} \subseteq \bar{R} \subseteq S$ with $\bar{N} \triangleleft S$ and $\operatorname{ann}_{S}(\bar{N})=0 .$ Conclude from Exercises 3,5, and 6 that $\mathbf{Q}_{\max }(R) \cong$ $\oplus \sum_{i} \mathbf{Q}_{\max }\left(R / P_{i}\right)$ in general and that $Q_{\mathrm{cl}}(R) \cong \oplus \sum_{i} \mathrm{Q}_{\mathrm{cl}}\left(R / P_{i}\right)$ under appropriate hypotheses.

If $R$ is a Dedekind domain, then the class group of $R$ is defined to be the multiplicative group of nonzero fractional ideals of $R$ modulo the subgroup of principal fractional ideals. For example, if $R$ is a principal ideal domain, then $\mathbf{C l}(R)=\{1\}$ and if $R$ is the ring of integers in an algebraic number field, then it is known that $\mathrm{Cl}(R)$ is finite.

Narayan Hari
Narayan Hari
Numerade Educator
02:33

Problem 9

Suppose $R$ is a Dedekind domain. If $A$ is a nonzero fractional ideal of $R$, let $\tilde{A}$ denote its natural image in $\mathbf{C l}(R) .$ Use Lemma $7.6$ and Theorem 7.7(i) to prove that $\mathbf{K}_{0}(R) \cong \mathbb{Z} \oplus \mathbf{C l}(R)$ via the map determined by $[A] \mapsto 1 \oplus \tilde{A} .$ In particular, conclude that $\mathbf{K}_{0}(R)=\langle[R]\rangle+H$ with $\langle[R]\rangle$ infinite cyclic and with $H \cong \mathbf{C l}(R) .$ What about the structure of $\mathbf{G}_{0}(R) ?$

Foster Wisusik
Foster Wisusik
Numerade Educator
07:44

Problem 10

Let $R$ be a Noetherian domain with division ring of fractions $D$. If $H$ is the kernel of the induced module map $\mathrm{G}_{0}(R) \rightarrow \mathbf{G}_{0}(D)$, prove that $\mathbf{G}_{0}(R)$ is generated by $[R]$ and $H$. Obtain an analogous result for $\mathbf{K}_{\mathbf{0}}(R)$.

Anthony Ramos
Anthony Ramos
Numerade Educator