Chapter 20, Nullstellensatz Video Solutions, A Course in Ring Theory

Problem 1

What difference would it make in the definition of an almost centralizing extension $R \subseteq S$, if we assumed that the centralizing generators $x_{1}, x_{2}, \ldots, x_{n}$ satisfied the stronger condition $x_{i} x_{j}-x_{j} x_{i} \in \sum_{k=1}^{n} R x_{k}$ for all $i, j$ ? This has a simple one-line answer.

Chareen Guzman

Numerade Educator

02:22

Problem 2

Let $\mathbb{Z}$ denote the ring of integers and let $T$ be the multiplicatively closed subset of $\mathbb{Z}$ generated by the primes $p_{1}, p_{2}, \ldots$ Then $R=\mathbb{Z} T^{-1}$ can be filtered by defining $R_{n}$ to be the set of all fractions of the form $z / p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{k}^{a_{k}}$ with $z \in \mathbb{Z}$ and all $a_{i} \leq n$. Prove that $\operatorname{gr} R \cong$ $\mathbb{Z} \oplus x S[x]$, where $S$ is the ring (without 1 ) given by $S=\oplus \sum_{i} \mathrm{GF}\left(p_{i}\right)$.

James Chok

Numerade Educator

01:23

Problem 3

Use the preceding exercise to construct a filtered domain $R$ such that $\operatorname{Rad}(R) \neq 0$ but $\operatorname{Rad}(\operatorname{gr} R)=0 .$ Conversely, construct a semiprimitive filtered commutative domain $R^{\prime}$ with $\mathrm{Nil}\left(\mathrm{gr} R^{\prime}\right) \neq 0$
$R \subseteq S$ is a finite normalizing extension of rings if $S=\sum_{i=1}^{n} y_{i} R$ for suitable $y_{i} \in S$ with $y_{i} R=R y_{i}$ for all $i$.

Hast Aggarwal

Numerade Educator

03:58

Problem 4

Let $R \subseteq S$ be a finite normalizing extension and let $V$ be an irreducible $S$-module. Prove that $V_{R}$ is a finite direct sum of irreducible $R-$ modules. If the extension is centralizing, show that all the irreducible $R$-summands of $V$ are isomorphic.

Anthony Ramos

Numerade Educator

02:26

Problem 5

Let $V$ be the $A_{m}(K)$-module described immediately preceding Theorem 20.8. Prove that $V$ is irreducible if $\operatorname{char} K=0$ but reducible otherwise.
The next three problems discuss Amitsur's simple proof of the Nullstellensatz over uncountable fields.

Chris Trentman

Numerade Educator

03:36

Problem 6

Let $R$ be an algebra over the field $K$ and let $r \in R$. If $1-k r$ is invertible for all $k \in K$ and if $\left\{(1-k r)^{-1} \mid k \in K\right\}$ is $K$-linearly dependent, prove that $r$ is algebraic over $K$. Furthermore, show that any $s \in$ $\operatorname{Rad}(R)$ which is algebraic over $K$ must necessarily be nilpotent.

Ahmad Reda

Numerade Educator

02:12

Problem 7

Let $R$ be a $K$-algebra and assume that $\operatorname{dim}_{K} R<|K|$, where this is an inequality of possibly infinite cardinals. Prove that $\operatorname{Rad}(R)$ is a nil ideal. Furthermore, if $R$ is a division ring, prove that $R$ is algebraic over $K$.

Linh Vu

Numerade Educator

13:07

Problem 8

Suppose now that $K$ is an uncountable field and that $R$ is a finitely generated $K$-algebra. Prove that $\operatorname{Rad}(R)$ is nil. Furthermore, if $V$ is an irreducible $R$-module, prove that the division ring $D=\operatorname{End}_{R}(V)$ is algebraic over $K$.

If $K$ is a countable field and $R$ is a finitely generated $K$-algebra, then $\operatorname{Rad}(R)$ need not be nil. Although counterexamples exist in all characteristics, we will only offer a characteristic 0 example here. Its construction is analogous to the construction of the Weyl algebras.
Let $C$ be a commutative algebra over the field $K$ and let $C[[\zeta]]$ be the power series ring over $C$ in the variable $\zeta .$ View $V=C[[\zeta]]$ as a $C$-module and consider End $_{C}(V)$, acting on the right. Since multiplication by any element of this ring is a $C$-endomorphism, we have an embedding of $C[[\zeta]]$ into $\operatorname{End}_{C}(V)$, say $C[[x]] \subseteq \operatorname{End}_{C}(V)$. In addition, let $y \in \operatorname{End}_{C}(V)$ denote the operator $\partial / \partial \zeta$ and let $e \in \operatorname{End}_{C}(V)$ be the map $v(\zeta) \mapsto v(0)$ that reads off the constant coefficient.

Oswaldo Jiménez

Numerade Educator

02:48

Problem 9

If $\alpha \in \operatorname{End}_{C}(V)$, prove that $e \alpha e=e c$, where $c=(1 \alpha)(0) \in C$ Conclude that $e$ is an idempotent and that $e \operatorname{End}_{C}(V) e=e C \cong C$, where the latter is a ring isomorphism. Next, if $\beta \in C[[x]]$, prove that $\beta y-y \beta=\partial \beta / \partial x .$

Mohan Jain

Numerade Educator

13:07

Problem 10

Now let $K$ be a countable field of characteristic $0 .$ Find a countable commutative $K$-algebra $C$ such that $C$ is a domain and $\operatorname{Rad}(C) \neq 0$. For example, take $C$ to be a suitable localization of a polynomial ring over $K$. Next, let $C=\left\{c_{0}, c_{1}, c_{2}, \ldots\right\}$ and define $\gamma \in \operatorname{End}_{C}(V)$ by $\gamma=\sum_{n=0}^{\infty} c_{n} x^{n} / n !$. If $R=K[e, \gamma, y]$ is the finitely generated $K$ subalgebra of $\operatorname{End}_{C}(V)$, prove that $e R e=e C .$ Deduce that $\operatorname{Rad}(R) \supseteq$ $\operatorname{Rad}(e R e) \cong \operatorname{Rad}(C)$ is not nil.

Oswaldo Jiménez

Numerade Educator