# Questions tagged [ra.rings-and-algebras]

Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.

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242 views

### Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$

Edit: According to the comment of Todd Trimble, I revise the question.
What are some examples of functors $F$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\cong ...

**0**

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**0**answers

101 views

### Homomorphisms from $k[x,y]$ to $k[x,x^{-1},y]$

Let $k$ be a field of characteristic zero and let $R_{-1}:=k[x,x^{-1},y]$
be the $k$-algebra of polynomials in $x,y$ containing the inverse of $x$, denoted by $x^{-1}$.
Let $f: k[x,y] \to R_{-1}$ be ...

**2**

votes

**0**answers

28 views

### Denominator identity for Lie superalgebras

Let $\mathfrak g$ be a basic classic simple Lie superalgebra.
Fix a maximal isotropic subset $S \subset \Delta$ and choose a set of simple roots $\Pi$ containing $S$. Let $R$ be the Weyl ...

**1**

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**0**answers

78 views

### Could we assume without loss of generality that all coefficients are positive?

Let $\alpha$ be an element in the group algebra $\mathbb CG$ of a torsion-free group $G$. Assume that, as an operator acting on $\ell^2(G)$, $\alpha$ is positive. Does there exist $\beta\in\mathbb CG$ ...

**5**

votes

**2**answers

395 views

### Zero tensor product over a complex algebra?

Let $A$ be an algebra over $\mathbb{C}$. Let $M$ be a left $A$-module, let $N$ be a right $A$-module and consider the tensor product $N \otimes_A M$, which is a complex vector space.
Q1: Can this ...

**2**

votes

**1**answer

90 views

### Name and properties of this combination of group algebra and semidirect product?

Given a field $k$, a group $G$, and a homomorphism $\phi : G \to \mathrm {Aut} (k)$, we can define a ring $\widehat {k [G]}_\phi$ as follows: As an abelian group it is isomorphic to the group algebra $...

**2**

votes

**1**answer

95 views

### Field of definition of a finite dimensional division algebra and how to reduce it

Let F be a field, and E/F an infinite algebraic extension. Let D be a finite dimensional division algebra over E (meaning its center is also E).
Is it possible to somehow gow down to a finite ...

**2**

votes

**1**answer

239 views

### When does the forgetful functor from modules to vector spaces have a right adjoint?

Given any algebra $R,$ when does the forgetful functor
$R\text{-}Mod \rightarrow Vec$
have a right adjoint?
Does this imply any finiteness conditions on R?
Is there a book/paper discussing this?
I'...

**3**

votes

**1**answer

77 views

### A “concrete” example of a one-sided Hopf algebra

I came to know from the paper Left Hopf Algebras by Green, Nichols and Taft that one may consider a Hopf algebra whose antipode satisfies only the left (resp. right) antipode condition.
To be more ...

**4**

votes

**0**answers

100 views

### Contractible Banach algebras

A Banach algebra $A$ is contractible if $H^1(A, X)=0$ for all Banach $A$-bimodules $X$. Now to my question
Let $A$ be Banach algebra and $I$ be closed ideal of $A$. If $I$ and $A/I$ are both ...

**8**

votes

**1**answer

236 views

### Example of a ring where every module of finite projective dimension is free?

I'm interested in seeing an example of a ring which is not self-injective where every module admitting a finite projective resolution is free, or at least projective.
Note that self-injectivity says ...

**3**

votes

**1**answer

122 views

### Multiplication in $Z(\mathbb{C}S_n)$ [duplicate]

I am trying to multiply two generators of center $Z(\mathbb{C}[S_n])$ of ring algebra of symmetric group of $n$ elements. We know that these generators are given by sums of conjugacy classes in $S_n,$ ...

**2**

votes

**0**answers

42 views

### Generalizing polynomial identities for rings

For a ring $R$, a polynomial identity of $R$ is a polynomial (in non-commuting variables) $f(x_1,\ldots,x_n)\in \mathbb{Z}[x_1,\ldots, x_n]$ such that for any choice of $a_i\in R$, $f(a_1,\ldots, a_n)=...

**6**

votes

**0**answers

154 views

### Homotopy quotient of groups

Suppose $0\to A \stackrel{\iota}{\to} B \stackrel{\pi}{\to} C \to 0$ is a short exact sequence of groups.
We have an induced map $k[\iota] : k[A] \to k[B]$ of group algebras over a field $k$.
What ...

**2**

votes

**0**answers

77 views

### Open sets on a Stone space

If $B$ is a Boolean algebra (possibly assumed complete), is there a standard name for the Heyting algebra (or frame) $L := \Omega(S(B))$ of open sets on the Stone space $S(B)$ of $B$, — or for the ...