2
votes
1answer
94 views

Does an ideal of finite codimension in a finitely generated algebra have always to be finitely generated?

I have been reading a book on Lie Algebras ("√Ālgebras de Lie" by San Martin) and there is this exercise in the chapter on universal enveloping algebras with a claim that I can not prove: Suppose ...
2
votes
1answer
50 views

If $S$ be a ring with identity and $f: M_{2}(\mathbb{R})\longrightarrow S$ is a nonzero ring homomorphism, which cases is true?

Let $S$ be a ring with identity and $f: M_{2}(\mathbb{R})\longrightarrow S$ is a nonzero ring homomorphism,then which cases is true? $S$ is left Artinian $S$ is left Noetherian $S$ is simple ring ...
6
votes
1answer
76 views

A finite unital and commutative ring with exactly one maximal ideal has $p^{n}$ elements.

Suppose $R$ is a finite unital and commutative ring that has exactly one maximal ideal. Prove that $\left | R \right |=p^{n}$ where $p$ is a prime number. If $R$ will be non-commutative, do we have ...
1
vote
1answer
91 views

Quotient ring of a matrix ring

Let $F$ be a field. Consider the set $$R = \left\{ \begin{bmatrix} a & b \\ 0 & c \end{bmatrix} : a, b, c \in F \right\}$$ Think of a non-trivial two-sided ideal of $R$ and describe in ...
0
votes
0answers
21 views

Abstract Algebra Question concerning $o(3)$ generators in a Hopf Algebra

Define $q = e^{\eta}$, such that $q$ is an arbitrary parameter. I need to show the following; $$0 = -q^{\mp 1} L_{\pm} q^{L_0} + q^{L_0} L_{\pm}$$ Where $L_{\pm}, L_0$ are the $o(3)$ generators. ...
2
votes
1answer
76 views

Quantum Hamiltonian commuting with the Pauli-Runge vector.

I have to prove that $[A_j, H] = 0$, with; $$\vec{A} = \frac{1}{2Ze^{2}m}(\vec{L} \times \vec{P} - \vec{p} \times \vec{L}) + \frac{\vec{r}}{r}$$ $$H = \frac{p^2}{2m} - \frac{Ze^2}{r}$$ And, $Z, e, ...
0
votes
1answer
89 views

Prove that a module is projective or not

Let $R=\left(\begin{array}{cc}\mathbb{Q}&\mathbb{Q}\\0&\mathbb{Q}\end{array}\right), J=\left(\begin{array}{cc}0&\mathbb{Q}\\0&0\end{array}\right)$. Prove that $R/J$ is not a ...
0
votes
0answers
64 views

Universal Enveloping Algebra and Commutation

I've been given the following question; Define elements $A^{m}_{ij}$ of the universal enveloping algebra $U(gl(n))$ recursively by $$A^{m}_{ij} = \sum^{n}_{k=1} a^{i}_{k} A^{m-1}_{kj}$$ ...
1
vote
1answer
79 views

A corollary of Koethe's theorem

A homework problem is divided into 2 parts: I managed to solve the first part, which states: Prove Koethe's theorem: If $D$ is a finite dimensional central division $k$-algebra and $K_0 \subset ...
5
votes
0answers
155 views

Problem 24 of section 2 of *Noncommutative Algebra* by Farb & Dennis

Problem 24 of section 2 of Noncommutative Algebra by Farb & Dennis states: Let $R$ be an artinian ring and let $G$ be a finite group. Show that $R[G]$ is semisimple if and only if $R$ is ...