Chapter 26, Goldie Rings Video Solutions, A Course in Ring Theory

Problem 1

Let $Q=\mathbf{Q}_{\max }(R)$ or $\mathrm{Q}_{\mathrm{r}}(R)$ or $\mathbf{Q}_{\mathrm{s}}(R)$ or $\mathbf{Q}_{\mathrm{cl}}(R) .$ If $I$ is a nonzero ideal of $Q$, prove that $I \cap R \neq 0$. Conclude that if $R$ is prime or semiprime, then the same is true of $Q$.

Sriparna Bhattacharjee

Numerade Educator

01:08

Problem 2

Let $K$ be a field. Show that the Artinian ring $R=\left(\begin{array}{cc}K & K \\ 0 & K\end{array}\right)$ is both right and left nonsingular. Conclude that $\mathbf{Q}_{\max }(R) \neq \mathbf{Q}_{\mathrm{cl}}(R)$. (See Exercise 24.5.)

Victor Salazar

Numerade Educator

02:45

Problem 3

Show that the right and left Noetherian ring $R=\left(\begin{array}{ll}\mathbb{Z} & \mathbb{Z} / 2 \mathbb{Z} \\ 0 & \mathbb{Z} / 2 \mathbb{Z}\end{array}\right)$ is right nonsingular but not left nonsingular. Find an element $d \in R$ with $\operatorname{lann}_{R}(d)=0$ but $\operatorname{r} \cdot \operatorname{ann}_{R}(d) \neq 0$

Prashant Bana

Numerade Educator

01:39

Problem 4

Let $W \subseteq V$ be $R$-modules. If $W$ and $V / W$ are nonsingular, prove that $V$ is nonsingular. Show that $(\operatorname{Sing}(V)+W) / W \subseteq \operatorname{Sing}(V / W)$ but that equality need not occur.

Nick Johnson

Numerade Educator

02:45

Problem 5

Show that $\operatorname{Sing}(R)$ contains no nonzero idempotents. Deduce that a von Neumann regular ring must be nonsingular. In particular, if $R$ is a nonsingular ring, conclude that $\mathbf{Q}_{\max }(R)$ is also nonsingular.

Prashant Bana

Numerade Educator

03:38

Problem 6

Prove that a commutative ring is nonsingular if and only if it is semiprime. If $R$ is the ring of continuous functions on the closed interval $[0,1]$, show that $\mathbf{Q}_{\max }(R)$ is a commutative von Neumann regular ring with no minimal ideals.

Prathan Jarupoonphol

Numerade Educator

01:05

Problem 7

If $V$ is an $R$-module and $V \supseteq W \supseteq \operatorname{Sing}(V)$, define $W^{*} \supseteq W$ by $W^{*} / W=\operatorname{Sing}(V / W) .$ Prove that $W$ ess $W^{*}$ and then use $W$ ess $W^{* *}$ and Lemma 25.10(i) to deduce that $W^{*}=W^{* *}$. In particular, if we define $\operatorname{Sing}_{2}(V)=\operatorname{Sing}(V)^{*}$, conclude that $V / \operatorname{Sing}_{2}(V)$ is nonsingular.

Anthony Ramos

Numerade Educator

03:56

Problem 8

Let $V=(\mathbb{Z} / 4 \mathbb{Z}, \mathbb{Z} / 4 \mathbb{Z})$ be the natural right module for the ring $R=$ $\left(\begin{array}{cc}\mathbb{Z} / 4 \mathbb{Z} & \mathbb{Z} / 4 \mathbb{Z} \\ 0 & \mathbb{Z} / 4 \mathbb{Z}\end{array}\right)$. Compute $\operatorname{Sing}(V)$ and $\operatorname{Sing}_{2}(V)$

Melvin Adkins

Numerade Educator

03:38

Problem 9

Assume that $\mathbf{Q}_{\mathrm{cl}}(R)$ exists. If every essential right ideal of $R$ contains a regular element, prove that $\mathbf{Q}_{\max }(R)=\mathbf{Q}_{\mathrm{cl}}(R)$. In particular, if $R$ is a semiprime Goldie ring, conclude that $\mathbf{Q}_{\max }(R)=\mathrm{Q}_{\mathrm{cl}}(R)$.

Prathan Jarupoonphol

Numerade Educator

01:05

Problem 10

Let $R$ be a semiprime Goldie ring and let $V$ be a right $R$-module. Assume that, for every $v \in V$ and regular element $t \in R$, there exists a unique $v^{\prime} \in V$ with $v^{\prime} t=v .$ Prove that $V$ is injective. In particular, deduce that every $\mathbf{Q}_{\mathrm{cl}}(R)$-module is injective as an $R$-module.

Anthony Ramos

Numerade Educator