Chapter 5, Artinian Rings Video Solutions, A Course in Ring Theory

Problem 1

If $f$ and $g$ are orthogonal idempotents, show that $e=f+g$ is an idempotent and that $e R=f R+g R$. Conclude that an idempotent
$e^{\prime}$ of $R$ is primitive if and only if $e^{\prime} R$ is indecomposable as a right $R$-module.

Urvashi Arora

Numerade Educator

04:55

Problem 2

Let $N$ be a nil ideal of $R$ and let $\left\{\bar{e}_{1}, \bar{e}_{2}, \ldots, \bar{e}_{n}\right\}$ be a set of orthogonal idempotents of $R / N$ that sum to $1 .$ If $\nu: R \rightarrow R / N$ is the natural epimorphism, prove that there exists a set $\left\{e_{1}, e_{2}, \ldots, e_{n}\right\}$ of orthogonal idempotents of $R$ with $\nu\left(e_{i}\right)=\bar{e}_{i}$ and $e_{1}+e_{2}+\cdots+e_{n}=1$. Furthermore, if $\left\{f_{1}, f_{2}, \ldots, f_{n}\right\}$ is a second such lifting, show that $u=e_{1} f_{1}+e_{2} f_{2}+\cdots+e_{n} f_{n}$ is a unit of $R$ with $u^{-1} e_{i} u=f_{i}$ for all $i$.
A ring $R$ is called semiprimary if $N=\operatorname{Nil}(R)$ is a nilpotent ideal with $R / N$ a Wedderburn ring. In particular, by Theorem 5.3, any Artinian ring is necessarily semiprimary. Furthermore, it is easy to see that the results of Proposition $5.5$ and Theorem $5.9$ apply to these more general rings.

Chris Trentman

Numerade Educator

00:15

Problem 3

Find an example of a semiprimary ring that is not Artinian. A suitable upper triangular $2 \times 2$ matrix ring will work.

Fuzail Shakir

Numerade Educator

04:10

Problem 4

Let $I$ be a nonnilpotent right ideal of the semiprimary ring $R$. Use Lemma $5.8$ to prove that $I$ contains a nonzero idempotent. Furthermore, show that $I$ is minimal with the property of being nonnilpotent if and only if $I=e R$ for some nonzero primitive idempotent $e \in R$. Notice that such right ideals correspond precisely to the projective indecomposables of $R$.

A ring $R$ is said to be von Neumann regular if every cyclic right ideal is generated by an idempotent.

Manisha Sarker

Numerade Educator

02:22

Problem 5

Prove that $R$ is von Neumann regular if and only if for all $r \in R$ there exists $r^{\prime} \in R$ with $r r^{\prime} r=r$. Conclude that the definition of von Neumann regular is right-left symmetric.

Ameer Said

Numerade Educator

12:56

Problem 6

Show that any Wedderburn ring is von Neumann regular. Conversely, if $R$ is von Neumann regular and Artinian, show that it is a Wedderburn ring. Give an example of a von Neumann regular ring that is not Wedderburn. For this, let $X$ be an infinite set, let $D$ be a division ring and consider the ring of all functions from $X$ to $D$ with pointwise addition and multiplication.

Chris Trentman

Numerade Educator

01:38

Problem 7

Let $I$ be a two-generator right ideal of the von Neumann regular ring $R$. Show that $I=e R+f R=e R+(1-e) f R=e R+g R$, where $e, f$ and $g$ are idempotents of $R$ with $e g=0$. Then show that $I=(1-g) e R+g R$ and observe that $g$ and $(1-g) e$ are orthogonal idempotents of $R$. Conclude that $h=(1-g) e+g$ is an idempotent and that $I=h R$. It follows that every finitely generated right ideal of $R$ is generated by an idempotent and hence is a direct summand of $R$.

Urvashi Arora

Numerade Educator

04:03

Problem 8

If $\theta: R \rightarrow S$ is a ring epimorphism, prove that $\theta(\operatorname{Nil}(R)) \subseteq \operatorname{Nil}(S)$ and that $\theta(\operatorname{Rad}(R)) \subseteq \operatorname{Rad}(S)$. Show by example that these inclusions need not be equalities. What happens if $\theta$ is not an epimorphism? Finally, find a commutative integral domain $R$ with $\operatorname{Rad}(R) \neq 0$. Conclude that $\operatorname{Rad}(R)$ can be properly larger than $\operatorname{Nil}(R)$.

Uma Kumari

Numerade Educator

01:00

Problem 9

Let $I$ be a right ideal of $R$ and let $A=\{r \in R \mid(R / I) r=0\}$. Prove that $A$ is the largest two-sided ideal of $R$ contained in $I$.

Jacquelyn Trost

Numerade Educator

01:16

Problem 10

Let $C=C\left(a_{1}, a_{2}, \ldots, a_{n}\right)$ be a Clifford algebra with $a_{i} \neq 0$ precisely when $i \leq r .$ Prove that $\operatorname{Rad}(C)$ is generated by $x_{r+1}, \ldots, x_{n}$ and that $C / \operatorname{Rad}(C) \cong C\left(a_{1}, \ldots, a_{r}\right)$

Amrita Bhasin

Numerade Educator