# ra.rings-and-algebras's questions - English 1answer

1.885 ra.rings-and-algebras questions.

### 3 indecomposable semi local Bezout ring

A commutative Bezout ring with $1$ is a ring in which every finitely generated ideal is principal. Is there any charactrization for indecomposable semi local Bezout ring? For example , for a ...

### 4 Group rings such that every (countable generated) module has a maximal submodule

Every (non-zero) finitely generated module over a ring has a maximal proper submodule by a simple application of Zorn's lemma. I am interested in the following question, with two variants. ...

### 2 Units in the (stable) center of a Frobenius algebra [duplicate]

Let $A$ be a Frobenius algebra with center $Z(A)$ and $I\subset Z(A)$ the ideal of elements in the image of some $A$-bimodule map $A\rightarrow A\otimes A\rightarrow A$, where the second map is ...

### 4 Values attained by the coheight of $(H \setminus H^\times)^k$ as a function of $H$ and $k$

Edit (Apr 24, 2017). I'm updating this post in the light of the latest developments of a related thread. Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$,...

### 4 Division ring on a field

Suppose that $F$ is a field. Show that there exists a $F$-division algebra $D$ with two elements $a\neq b\in D$ such that $a^2-2ab+b^2=0$. In the field extensions we know that $a^2-2ab+b^2=0$ if and ...

### 7 Can a positive polynomial on sphere be represented as the sum of squares of spherical harmonics

Let $p\in {\mathbb{R}}[x_1,\ldots, x_d]$ be a homogenous polynomial degree $2n$. We know that if $p$ is positive on $[-\pi,\pi]^d$, $p$ is sum of squares polynomial, i.e. $p$ can be witten as sum of ...

### Relation between left projections

Let $A$ be a Baer *-ring. Let $x$ be in $A$, $L(x)$ is the left projection of $x$ that is the smallest projection with $L(x)x=x$. Q. Let $p,q$ are projections in $A$ with $p\leq q$. For a given ...

### 5 Center of $k$-algebra with two generators and sole defining relation $yx - xy = 1$ when $\text{char}\,k > 0$

Let $A(k)$ be a $k$-algebra with two generators, $x$, $y$, and one defining relation: $yx - xy = 1$. What is the center of the algebra $A(k)$ in the case $\text{char}\,k > 0$?

### 24 Is there a field $F$ which is isomorphic to $F(X,Y)$ but not to $F(X)$?

0 answers, 760 views ra.rings-and-algebras examples fields

### 1 Derivations of special rings

Consider $p$-adic field $\mathbb{Q}_p$ and let $A$ be any finite dimensional $\mathbb{Q}_p$ algebra without zero divisors (and maybe without $1$). Now we can look on $A$ as on a ring $A_{\mathbb{Z}}$ (...

### 3 Ideals and Idempotents in a commutative ring

Let $I$, $J$, and $K$ be pairwise comaximal ideals of a commutative ring $R$ with $1$ with the property that if $x^2-x\in I$, then there exists $e^2=e\in J$ such that $x-e\in I$ and if $y^2-y\in K$, ...

### 1 Relation between coefficients of expansions

Related to Relations between coefficients of expansions of a rational function at 0 and infinity I commented at the linked question that the question seemed less about what happened "at infinity", ...

### 3 How to show that the graded dual of the universal enveloping algebra of a free Lie algebra on a finite set is the shuffle algebra

In the article, the universal enveloping algebra of a free Lie algebra on a set X is defined to be the free associative algebra generated by X. It is said that the graded dual of the universal ...

### 9 Why should we study derivations of algebras?

3 answers, 881 views ra.rings-and-algebras derivations
Some authors have written that derivation of an algebra is an important tools for studying its structure. Could you give me a specific example of how a derivation gives insight into an algebra's ...

### 3 Cancellation property for certain integral group rings

We say that the cancellation property holds for a ring $R$ if for finitely generated projective $R$-modules $P$ and $Q$ we have that $R \oplus P \cong R \oplus Q$ implies $P \cong Q$. In the case ...

### 1 Examples of reduced associative algebras

An associative $K$-algebra A is called reduced (or often basic) if $A/rad(A)$ has no nilpotent elements. It can be shown that this is equivalent to that $A/rad(A)$ is a isomorphic to a direct sum of ...

### 4 Question on $n$-torsionless modules

Let $A$ be a finite dimensional algebra. Recall that a module $M$ is called $n$-torsionfree in case $Ext_A^i(D(A),\tau(M))=0$ for all $i=1,...,n$ when $\tau$ denotes the Auslander-Reiten translate. ...

### 2 generalisations of module maps

For an algebra $A$ over a field $\mathbb{K}$, consider left modules $E,F$. The two obvious collections of maps to take from $E$ to $F$ are the left module maps ${}_A\mathrm{Hom}(E,F)$ and the linear ...

### 1 Strongly finite projections in $*$-rings

Let $A$ be a $*$-ring. Let us have some points: i) We recall that a projection $p$ is a self-adjoint idempotent that is $p=p^*=p^2$. ii) On the set of projections, we write $p\leq q$ if $pq=p$. iii)...

### 4 Central idempotents from characters in Frobenius algebras (generalizing Lusztig arXiv:math/0208154v2 §19)

$\newcommand{\refone}{\textbf{(1)}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\tr}{\operatorname{Tr}} \newcommand{\kk}{\mathbf{k}}$ Let $\kk$ be a field. Let $A$ be a $\kk$-algebra which is ...

### 11 About enveloping algebras of direct sums

3 answers, 578 views ra.rings-and-algebras lie-algebras
This question is imported from MSE. It is linked to this one in the case of semi-direct products. My question Let us consider a Lie $R$-algebra ($R$ is a commutative ring) written as a (module) ...

### 5 Counterexample for the Skolem-Noether Theorem

1 answers, 205 views ra.rings-and-algebras division-rings
If a division ring is finite-dimensional over its center then we can apply Skolem-Noether theorem (which asserts that every endomorphism is a conjugation). Can someone give a counterexample of the ...

### 1 Example of tensor category with non-simple unit $J\to \mathbb{1} \to Q$ and suitably extension $Q\to M\to J$

Edit: Thanx very much to Neil Strickland for quickly explaining to us that the following cannot be realized over finite commutative $\mathbb{C}$-algebras, as I had originally asked. I know that there ...