Chapter 29, Reduced Rank Video Solutions, A Course in Ring Theory

Problem 1

If $S$ is a ring with one of the following properties, show that $S$ is not embeddable in an Artinian ring.
i. $S$ has arbitrarily large finite subsets of orthogonal idempotents. ii. $S$ does not satisfy the maximal or minimal condition on right annihilators of subsets.
iii. There exist infinitely many rational primes $p$ with $\operatorname{ann}_{S}(p) \neq 0$

Nick Johnson

Numerade Educator

Problem 2

Suppose $S$ is a subring of the Artinian ring $Q .$ If $V$ is a finitely generated $S$-module, define $\sigma(V)=\operatorname{len}_{Q}\left(V \otimes_{S} Q\right) / \operatorname{len}_{Q}(Q) .$ Show that
i. $\sigma(V)$ is a nonnegative rational number with $\sigma(S)=1$.
ii. $\sigma\left(V_{1} \oplus V_{2}\right)=\sigma\left(V_{1}\right)+\sigma\left(V_{2}\right)$.
iii. If $U \rightarrow V \rightarrow W \rightarrow 0$ is an exact sequence, then $\sigma(W) \leq \sigma(V) \leq$ $\sigma(U)+\sigma(W)$
Such a function $\sigma$ is called a Sylvester rank function. Amazingly, Schofield has proved that every algebra with a Sylvester rank function is embeddable in an Artinian ring.

Check back soon!

03:13

Problem 3

Let $X$ be a subset of the ring $S$ and assume that $S$ is contained in a right Artinian ring or in a left Noetherian ring. Prove that there exists a finite subset $X_{0} \subseteq X$ with r.ann $_{S}(X)=\mathrm{r} \cdot \operatorname{ann}_{S}\left(X_{0}\right)$.

Gideon Idumah

Numerade Educator

01:05

Problem 4

Let $e$ and $f$ be idempotents in a ring $S .$ Suppose first that $e=u v$ and $f=v u$ for some $u, v \in S$ and define $\alpha: e S \rightarrow S$ by $\alpha(e s)=v e s .$ Prove shat $\alpha$ determines an $S$-isomorphism from $e S$ to $f S .$ Conversely,

Anthony Ramos

Numerade Educator

04:00

Problem 5

Suppose $R$ is a right Noetherian ring containing the Wedderburn ring $S$ and write $S=\cdot \sum_{i=1}^{k} S_{i}$, a direct sum of simple rings. If $f_{i}$ is a primitive idempotent of $S_{i}$, prove that
$$
\text { u. } \operatorname{dim}_{R} R=\sum_{i=1}^{k}\left(\mathrm{u} \cdot \operatorname{dim}_{S} S_{i}\right)\left(\mathrm{u} \cdot \operatorname{dim}_{R} f_{i} R\right)
$$
To this end, first observe that $S_{i} \cong \mathrm{M}_{n_{i}}\left(D_{i}\right)$ has a family of orthogonal primitive idempotents $\left\{f_{i, 1}, f_{i, 2}, \ldots, f_{i, n_{i}}\right\}$ with $n_{i}=$ u.dim $_{S} S_{i}$ and $f_{i, j} S_{S} \cong f_{i} S_{S} .$ Then note that $\left\{f_{i, j}\right\}$ is an orthogonal decomposition of 1 in $R$ and hence that $R=\cdot \sum_{i, j} f_{i, j} R$. Finally, use the preceding exercise to conclude that $f_{i, j} R_{R} \cong f_{i} R_{R}$. This is a special case of the additivity principle.

Mohan Jain

Numerade Educator

03:56

Problem 6

Let $R$ be the ring $R=\left(\begin{array}{cc}\mathbb{Z} & \mathbb{Z} / p \mathbb{Z} \\ 0 & \mathbb{Z} / p \mathbb{Z}\end{array}\right)$, where $p$ is any prime. Show that $R$ is right and left Noetherian and that $N=\operatorname{Nil}(R)=\left(\begin{array}{cc}0 & \mathbb{Z} / p \mathbb{Z} \\ 0 & 0\end{array}\right)$ Prove that $\mathcal{C}(N) \neq \mathcal{C}(0)$ and that the right and left reduced ranks of the $(R, R)$-bimodule $N$ are distinct.

Sriparna Bhattacharjee

Numerade Educator

04:33

Problem 7

Suppose $A=A_{1}(K)$ is the first Weyl algebra over a field $K$ of characteristic 0 and let $0 \subset B \subset A$ be a proper right ideal of $A$. Note that $A / B=\operatorname{Sing}(A / B)$, since $B$ ess $A$, and that $\mathrm{r} \operatorname{ann}_{A}(A / B)=0$, since this annihilator must be a proper two-sided ideal of $A$. Now define $R=\left(\begin{array}{cc}K & A / B \\ 0 & A\end{array}\right)$ and observe that $R$ has a natural ring structure and as such it is a right Noetherian ring. If $X=\left(\begin{array}{cc}0 & A / B \\ 0 & 0\end{array}\right)$, show that every finite subset $X_{0}$ of $X$ is annihilated by some $\left(\begin{array}{ll}0 & 0 \\ 0 & E\end{array}\right)$ with $E$ ess $A$. Conclude from Exercise 3 that $R$ cannot be embedded in a right Artinian ring or a left Noetherian ring.
In the remaining problems, let $R$ denote a commutative Noetherian ring.

Lucía Guerrero

Numerade Educator

01:59

Problem 8

Let $P$ be a prime ideal of $R$. For each integer $n \geq 1$, show that there exists a unique $P$-primary component for the ideal $P^{n}$. We denote this component by $P^{(n)}$ and call it the $n$th symbolic power of $P .$ Suppose in addition that $R$ is an integral domain and set $R_{P}=R T^{-1}$, where $T=R \backslash P$. If $P^{\prime}=P R_{P}$, prove that $\left(P^{\prime}\right)^{n} \cap R=P^{(n)}$

James Chok

Numerade Educator

07:08

Problem 9

If $R_{R}$ is a uniform module, use Lemma $26.3$ to show that every element of $R$ is either nilpotent or regular. Conclude that every element of $Q=Q_{c l}(R)$ is either nilpotent or invertible and hence that $Q$ is a semiprimary ring. Deduce that $Q$ is also Artinian.

Tim Strang

Numerade Educator

01:33

Problem 10

Now apply Lemma $28.6$ and the previous problem to conclude that any commutative Noetherian ring can be embedded in a commutative Artinian ring. Find an example of a commutative ring that cannot be embedded in an Artinian ring.

Amy Jiang

Numerade Educator